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PLANE
AND SPHERICAL
TRIGONOMETRY
PLANE AND SPHERICAL
TRIGONOMETRY
BOOKS BY
C. I. PALMER
(Published
by McGraw-Hill
Book
Company,
Inc.)
PALMER'S
Practical Mathematics'
Part I-Arithmetic
with Applications
Part II-Algebra
with Applications
Part III-Geometry
with Applications
Part IV-Trigonometry
and Logarithms
PALMER'S
Practical
Mathematics
PALMER'S
Practical
Calculus
PALMER
Plane
for Home
Trigonometry
Professor
Emeritus
WILBER
of Analytic
Mechanics,
Author of Practical
LEIGH
Armour Institute
1\1echanics
of Technology,
with Tables
Geometry
AND MISER'S
Algebra
hy Scott,
TAYLOR,
Foresman
and
Company)
AND FARNUM'S
Geometry
Solid Geometry
FOURTH
PALMER,
TAYLOR,
of
AND
Study
Study
PALMER
AND KRATHWOHL'S
(PuhliRherl
PALMER,
IRWIN
and Dean of Students,
Armour Institute
of a Series of Mathematics
Texts
AND LEIGH'S
Analytic
PALMER
College
CLAUDE
Late Professor of Mathematics
Technology;
Author
CHARLES
Plane and Spherical
PALMER
for Home
BY
EDITION
AND FARNUM'S
NINTH
Plane and Solid Geometry
IMPRESSION
,
In the earlier editions of Practical Mathematics, Geometry with Applications was Part II and Algebra with Applications was Part III. The Parts have bcen rearranged in
response to many requests from users of the book.
McGRAW-HILL
NEW
BOOK COMPANY,
YORK
AND
1934
LONDON
INC.
l
PREFACE
COPYRIGHT,
1914,
MCGRAW-HILL
PRINTED
IN THE
1916,
1925,
1934, BY THE
BOOK COMPANY, INC.
UNITED
STATES
OF AMERICA
All rights reserved. This book, or
parts thereof, may not be reproduced
in any form without permission of
the publishers.
THE
MAPLE
PRESS
COMPANY,
YORK,
TO THE FOURTH EDITION
This edition presents a new set of problems in Plane Trigonometry.
The type of problem has been preserved, but the
details have been changed.
The undersigned acknowledges
indebtedness to the members of the Department of Mathematics
at the Armour Institute of Technology for valuable suggestions
and criticisms.
He is especially indebted to Profs. S. F. Bibb
and W. A. Spencer for their contribution of many new identities
and equations and also expresses thanks to Mr. Clark Palmer,
son of the late Dean Palmer, for assisting in checking answers to
problems and in proofreading and for offering many constructive
criticisms.
CHARLES
CHICAGO,
June, 1934.
PA.
v
WILBER
LEIGH.
r
I
PREFACE
TO THE FIRST EDITION
This text has been written because the authors felt the need of
a treatment of trigonometry that duly emphasized those parts
necessary to a proper understanding
of the courses taken in
schools of technology.
Yet it is hoped that teachers of mathematics in classical colleges and universities as well will find it
suited to their needs. It is useless to claim any great originality
in treatment or in the selection of subject matter.
No attempt
has been made to be novel only; but the best ideas and treatment
have been used, no matter how often they have appeared in other
works on trigonometry.
The following points are to be especially noted:
(1) The measurement of angles is considered at the beginning.
(2) The trigonometric functions are defined at once for any
angle, then specialized for the acute angle; not first defined for
acute angles, then for obtuse angles, and then for general angles.
To do this, use is made of Cartesian coordinates, which are now
almost universally taught in elementary algebra.
(3) The treatment of triangles comes in its natural and logical
unler and is not JOfced to the first pages 01 the book.
(4) Considerable use is made of the line representation of the
trigonometric functions.
This makes the proof of certain theorems easier of comprehension and lends itself to many useful
applications.
(5) Trigonometric equations are introduced early and used
often.
(6) Anti-trigonometric
functions are used throughout the
work, not placed in a short chapter at the close. They are used
in the solutions of equations and triangles.
Much stress is laid
upon the principal values of anti-trigonometric
functions as used
later in the more advanced subjects of mathematics.
(7) A limited use is made of the so-called "laboratory
method" to impress upon the student certain fundamental ideas.
(8) Numerous carefully graded practical problems are given
and an abundance of drill exercises.
(9) There is a chapter on complex numbers, series, and hyperbolic functions.
vii
''
I
'
,'/
,
I
!
l
viii
PREFACE TO THE FIRST
EDITION
(10) A very complete treatment is given on the use of logarithmic and trigonometric tables. This is printed in connection with
the tables, and so does not break up the continuity of the trigonometry proper.
(11) The tables are carefully compiled and are based upon
those of Gauss. Particular attention has been given to the
determination of angles near 0 and 90°, and to the functions of
such angles. The tables are printed in an unshaded type, and the
arrangement on the pages has received careful study.
The authors take this opportunity to express their indebtedness to Prof. D. F. Campbell of the Armour Institute of Technology, Prof. N. C. Riggs of the Carnegie Institute of Technology,
and Prof. W. B. Carver of Cornell University, who have read
the work in manuscript and proof and have made many valuable
suggestions and criticisms.
THE AUTHORS.
CHICAGO,
September,
1914.
CONTENTS
PAGE
PREFACETOTHE FOURTHEDITION. . . . . . . . . . . . . . ..
V
PREFACETOTHE FIRST EDITION. . . . . . . . . . . . . . . . . vii
CHAPTER
I
INTRODUCTION
ART.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
Introductory remarks. . . . . . . . . . .
Angles, definitions.
. . . . . . . . .
Quadrants. . . . . . . . . . . . . . . .
Graphical addition and subtraction of angles.
Angle measurement. . . . . . . . . . . .
The radian.
. . . . . . . . . . . . . .
Relations between radian and degree. . . .
Relations between angle, arc, and radius.
.
Area of circular sector.
. . . . . . . . .
General angles. . . . . . . . . . . . . .
Directed lines and segments. . . . . . . .
Rectangular coordinates. . . . . . . . . .
Polar coordinates. . . . . . . . . . . . .
. .
.
. .
. .
. .
. .
. .
. .
. .
. .
. .
. .
. .
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1
2
3
3
4
5
6
8
10
12
13
14
15
CHAPTER II
TRIGONOMETRIC
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
FUNCTIONS
OF ONE ANGLE
Functions of an angle. . . . . . . . . . . . . . .
Trigonometric
ratios.
. . . . . . . . . . . . . . . . .
Correspondence
between angles and trigonometric
ratios.
.
Signs of the trigonometric
functions.
. . . . . . .
Calculation from measurements.
. . . . . . . . . . . .
Calculations from geometric relations.
. . . . . . . . . .
Trigonometric
functions of 30°. ..
............
Trigonometric
functions of 45°. . . .
. . . . . . .
Trigonometric
functions of 120° . .
. . . . . . . .
Trigonometric
functions of 0° .
. . . . . . . . . .
Trigonometric
functions cf 90°. . . . . . . . . . . . . .
Exponents of trigonometric
functions.
. . . . . . . . . .
Given the function of an angle, to construct the angle.
. .
Trigonometric
functions applied to right triangles.
. . . .
Relations between the functions of complementary
angles.
Given the function of an angle in any quadrant, to construct
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
17
17
18
19
20
21
21
22
22
23
23
25
26
28
30
the
angle. . . . . . . . . . . . . . . . . . . . . . . . . . 31
ix
r
x
ART.
CONTENTS
CONTENTS
CHAPTER III
RELATIONS BETWEEN TRIGONOMETRIC FUNCTIONS
30. Fundamental
relations between the functions of an angle.
. .
31. To express one function in terms of each of the other functions.
32. To express all the functions of an angle in terms of one functioI) of
the angle, by means of a triangle.
. . . .
. . . . .
33. Transformation
of trigonometric
expressions.
34. Identities.
. . . . . . . . . . . . . . . . . . . . .. .. .. ..
35. Inverse trigonometric
functions.
. . . . . . . . . . .
36. Trigonometric
equations.
. .
. . . . . . . . . .
General statement.
.
Solution of a triangle.
.
The graphical solution.
. .
The solution of right triangles
Steps in the solution.
. . .
Remark on logarithms.
. .
Solution of right triangles by
Definitions.
. . . . . . .
34
36
V
FUNCTIONS
OF LARGE ANGLES
46. Functions of !71' - e in terms of functions of e. . . . . . . . .
47. FUilptioni' of:
+ e in tnJJJ.j
vi iUlldiuns
of u. . . . . .
48. Functions of 71'- e in terms of functions of e .
49. Functions of 71'+ e in terms of functions of e .
50. Functions of ~71'- e in terms of functions of e.
51. Functions of !71' + e in terms of functions of e.
52. Functions of - e or 271'- e in terms of functions
53. Functions of an angle greater than 271'. . . . .
54. Summary of the reduction formulas. . . . . .
55. Solution of trigonometric equations. . . . . .
. . .
. . .
. . .
. . .
of e.
. . .
. . .
. . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
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.
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.
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.
.
.
.
.
.
.
.
.
.
62
63
63
64
65
65
66
67
67
71
PRACTICAL
66.
67.
68.
69.
70.
71.
72.
73.
74.
75.
76.
77.
.
.
.
x
.
.
.
.
.
.
.
.
. . . . . . . . . . PAGE
. . . . . . . . . . 85
. . . . . . . . . . 86
87
..
87
"""
AND RELATED
Accuracy.
. . . . . . . . . .
Tests of accuracy.
. . . . . . .
Orthogonal projection.
. . . . .
Vectors.
. . . . . . . . . . .
Distance and dip of the horizon.
Areas of sector and segment.
. .
Widening of pavements on curves
Reflection of a ray of light.
. .
Refraction of a ray of light.
. .
Relation between sin e, e, and tan
Side opposite small angle given.
Lengths of long sides given..
FUNCTIONS
INVOLVING
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
PROBLEMS
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
90
91
92
""". . . . .
93
95
"
. . .
99
97
. . . . 102
102
'"
""'"
for small angles.
. . . . . 103
. . . . . . . . . . . . . 105
.
"
e,
.
105
VIII
MORE
THAN
ONE ANGLE
78. Addition and subtraction
formulas.
. . . . . . . . . . . . . 108
79. Derivation of formulas for sine and cosine of the sum of two angles 108
80. Derivation of the formulas for sine and cosine of the difference of
two angles.
. . . . . . . . . . . . . .
. . . 100
01. .Pruof of the addition formulas for other values of the angles. . . 110
82. Proof of the subtraction
formulas for other values of the angles.
110
83. Formulas for the tangents of the sum and the difference of two
angles.
. . . . . . . . . . . . . . . . . . . . . . . .113
84. Functions of an angle in terms of functions of half the angle. . . 114
85. Functions of an angle in terms of functions of twice the angle.
. 117
86. Sum and difference of two like' trigonometric
functions
as a
product.
. . . . . . . . . . . . . . . . . . . . . . . . 119
87. To change the product of functions of angles to a sum.
. . . . 122
88. Important
trigonometric
series. . . . . . . . . . . . . . . . 123
CHAPTER IX
.
angle
" increases
. . . . . . .
76
. . . . . . .
78
79
80
82
83
. . . . . . .
. . . . . . .
. . . . . .
APPLICATIONS
CHAPTER
CHAPTER
VI
GRAPHICAL REPRESENTATION OF TRIGONOMETRIC
FUNCTIONS
56. Line representation
of the trigonometric
functions.
57. Changes in the value of the sine and cosine as the
from 0 to 3600. . . . . . . . . . . . . . .
58. Graph of y
= sin e. . . . . . . . . . .. .. .. ..
59. Periodic functions and periodic curves.
60. Mechanical construction of graph of sin e. . . .
61. Projection of point having uniform circular motion.
Summary. . . . . . .
. .
Simple harmonic motion.
. . . .
Inverse functions. . . . . . . . .
Graph of y = sin-l x, or y
= arc sin
CHAPTER VII
37
38
40
42
43
. . . . . . . . . . 47
. . . . . . . . . .
47
. . . . . . . . . . . . . . .
48
by
computation.
. . . . . . . . 48
. . . . . .
. . . . . . .
. . . . . . . . . . . . . . . . 49
logarithmic functions.
. . . . . . 54
. . . . . . . . . . . . . . . . 54
56
CHAPTER
62.
63.
64.
65.
PAGE
CHAPTER IV
RIGHT TRIANGLES
37.
38.
39.
40.
41.
42.
43.
44.
xi
ART.
89.
90.
91.
92.
93.
OBLIQUE TRIANGLES
General statement.
. . . . . . . . . . .
. . . . . .
Law of sines. . . . . . . . . . . . . . . . . . . . . . . .
Law of cosines. . . . . . . . . . . . . . . . . . . . . . .
Case 1. The solution of a triangle when one side and two angles
are given.
. . . . . . . . . . . . . .
.
.
Case II.
The solution of a triangle when two "sides and an angle
'.
opposite one of them are given. . . . . .
. . . . . .
130
130
132
132
136
xii
CONTENTS
ART.
94. Case III.
The solution of a triangle when two sides and tne PAGE
included angle are given.
First niethod.
..
140
95. Case III.
Second method.
. . . . . . ..
""
140
96. Case IV. The solution of a triangle when the three sides
are
given
143
""
97. Case IV. Formulas adapted to the use of logarithms.
. . . . . 144
CHAPTER
CHAPTER X
MISCELLANEOUS
TRIGONOMETRIC
EQUATIONS
98. Types of equations.
. . . . . . . . . . . . . . . . . . . .
99. To solve r sin 0 + 8 cos 0
t for 0 when r, 8, and t are known.
100. Equations in the form p sin =
a cos fJ
a, p sin a sin fJ
c, where p, a, and fJ are variables. = . . . . . . . = . b,. p cos
. . a . =.
101. Equations in the form sin (a
c sin a, where fJ and care
+ fJ)
known.
. . . . . . . . . . . =. . . . . . . . . . . . .
102. .Equations in the form tan (a
+ fJ)
known.
. . . . . . . . . . . =. c. tan
. . a,. where
. . . fJ. and
. . care
. .
103.
Equations
angles.
of the
form
t
=
0
+
sin t, where
0 and",
SPHERICAL
158
160
161
161
162
are given
. . . . . . . . . '". . . . . . . . . . . . . . . 162
CHAPTER XI
COMPLEX
NUMBERS,
CONTENTS
DEMOIVRE'S
THEOREM,
SERIES
104. Imaginary numbers.
. . . . . . . . . . . . . . . . . . . . 165
105. Square root of a negative number.
. . . . . . . . . . . . . 165
106. Operations with imaginary numbers.
. . . . . . . . . . . . 166
107. Complex numhers . . . . . . . . . . . . . . . . . . . . . 166
108. Conjugate complex numbers.
. . . . . . . . . . . . . . . . 167
109. Graphical representation
of ('ompJex nurnncr:"
Wi
110. Powers of i . . . . . . . . . . . . . . . . . . . . . . . . 169
111. Operations on complex numbers.
. . . . . . . . . . .
169
112. Properties of complex numbers.
. . . . . . . . . . . . . . .171
"
113. Complex numbers and vectors.
. . . . . . . . . . . . . . . 171
114. Polar form of complex numbers.
. . . . . . . . . . . . . . 172
115. Graphical representation
of addition.
. . . . . . . . . . . . 174
116. Graphical representation
of subtraction.
. . . . . . . . . . . 175
117. Multiplication
of complex numbers in polar form. . . . . . . . 176
118. Graphical representation
of multiplication.
. . . . . . . . . . 176
119. Division of complex numbers in polar form.
. . . . . .
176
120. Graphical representation
of division.
. . . . . . . .
"
121. In volution of complex numbers.
. . . . . . . . . . .'. . . . 177
122. DeMoivre's theorem for negative and fractional exponents.
. 178
123. Evolution of complex numbers.
. . . . . . . . . . . . ..
179
124. Expansion of sin nO and cos nO. . . . . . . . . . . .
182
125. Computation
of trigonometric
functions.
. . . . . . . . . . .. 184
"
126. Exponential
values of sin 0, cos 0, and tan O. . . . . . . . . . 184
127. Series for sinn 0 and cosn 0 in terms of sines or cosines of multiples
of O. . . . . . . . . . . . . . . . . . . . . . . . . . . 185
] 28. Hyperbolic functions.
. . . . . . . .
187
xiii
ART.
PAGE
129. Relations between the hyperbolic functions.
. . . . . . . . . 188
130. Relations between the trigonometric
and the hyperbolic functions
188
131. Expression for sinh x and cosh x in a series.
Computation
189
131'. Forces and velocities represented as complex numbers
189
XII
TRIGONOMETRY
132. Great circle, small circle, axis. . . . . . . . . . . . . . . . 193
133. Spherical triangle.
. . . . . . . . . . . . . . . . . . . . 193
134. Polar triangles. . . . . . . . . . . . . . . . . . . . . . . 194
135. Right spherical triangle. . . . . . . . . . . . . . . . . . . 195
136. Derivation of formulas for right spherical triangles. . . . . . . 196
137. Napier's rules of circular parts. . . . . . . . . . . . . . . . 197
138. Species. . . . . . . . . . . . . . . . . . . . . . . . . . 198
139. Solution of right spherical triangles. . . . . . . . . . . . . . 198
140. Isosceles spherical triangles.
. . . . . . . . . . . . . . . . 200
141. Quadrantal triangles.
. . . . . . . . . . . . . . . . . . . 201
142. Sine theorem (law of sines) . . . . . . . . . . . . . . . . . 202
143. Cosine theorem (law of cosines) . . . . . . . . . . . . . . . 202
144. Theorem.
. . . . . . . . . . . . . . . . . . . . . . . . 204
145. Given the three sides to find the angles. . . . . . . . . . . . 204
146. Given the three angles to find the sides. . . . . . . . . . . . 205
147. Napier's analogies.
. . . . . . . . . . . . . . . . . . . . 206
148. Gauss's equations.
. . . . . . . . . . . . . . . . . . . . 208
149. Rules for species in oblique spherical triangles. . . . . . . . . 209
150. Cases. . . . . . . . . . . . . . . . . . . . . . . . . . . 210
151. Case I. Given the three sides to find the three an~le:" .
211
152. Case 11. Given the three angles to find the thrce sides. . . . . 212
153. Case III. Given two sides and the included angle. . . . . . . 212
154. Case IV. Given two angles and the included side. . . . . . . 213
155. Case V. Given two sides and the angle opposite one of them. . 213
156. Case VI. Given two angles and the side opposite one of them. . 215
157. Area of a spherical triangle.
. . . . . . . . . . . . . . . . 215
158. L'Huilier's formula. . . . . . . . . . . . . . . . . . . . . 216
159. Definitions and notations.
. . . . . . . . . . . . . . . . . 217
160. The terrestrial triangle.
. . . . . . . . . . . . . . . . . . 217
161. Applications to astronomy. . . . . . . . . . . . . . . . . . 218
162. Fundamental points, circles of reference. . . . . . . . . . . . 219
Summary of formulas. . . . . .
. . . . . . . . . . . 222
Useful constants. . . . . . . . . . . . . . . . . . . . . . 225
INDEX. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
The contents for the Logarithmic and Trigonometric Tables
and Explanatory Chapter is printed with the tables.
xiv
GREEK
A, a.
B, (3.
r, 'Y.
.1, O.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Alpha
Beta
Gamma
Delta
E, E.
Z, t.
H, 'T/.
EI, ().
I, L .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Epsilon
Zeta
Eta
Theta
Iota
K, K. . . . . . . . Kappa
A, A. . . . . . . . Lambda
M,}J.. . . . . . . . Mu
--------
ALPHABET
N, II. .
Z, ~. .
0, o. .
IT, 7r. .
-
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Nu
Xi
Omicron
Pi
P, p. . . . . . . . Rho
1:, ()". . . . . . . . Sigma
T, T. . . . . . . . Tau
T, u. . . . . . . . Upsilon
<P,cf>. . . . . . . . Phi
X, x. . . . . . . . Chi
'1', if;. . . . . . . . Psi
11, . . . . . . . Omega
"'.
---------
I
l
CONTENTS
I
.
.
.
PLANE AND SPHERICAL
TRIGONOMErrRY
CHAPTER I
INTRODUCTION
GEOMETRY
1. Introductory remarks.-The
word trigonometry is derived
from two Greek words, TPL'Y"'IIOIl
(trigonon), meaning triangle, and
}J.ETpLa
(metria), meaning measurement.
While the derivation of
the word would seem to confine the subject to triangles, the
measurement of triangles is merely a part of the general subject
which includes many other investigations involving angles.
Trigonometry
is both geometric and algebraic in nature.
Historically, trigonometry developed in connection with astronomy, where distances that could not be measured directly were
computed by means oLaJlgltltLlicQ<il111e;:L1hat_~9JJld_be
measill'gd.
The beginning of these methods may be traced to Babylon and
Ancient Egypt.
The noted Greek astronomer Hipparchus is often called the
founder of trigonometry.
He did his chief work between 146 and
126 B. C. and developed trigonometry as an aid in measuring
angles and lines in connection with astronomy.
The subject of
trigonometry was separated from astronomy and established as
a distinct branch of mathematics by the great mathematician
Leonhard Euler, who lived from 1707 to 1783.
To pursue the subject of trigonometry successfully, the student
should know the subjects usually treated in algebra up to and
including quadratic equations, and be familiar with plane geometry, especially the theorems on triangles and circles.
Frequent use is made of the protractor, compasses, and the
straightedge in constructing figures.
While parts of trigonometry can be applied at once to the
solution of various interesting and practical problems, much of
1
'2
PLANE AND SPHERICAL
INTRODUCTION
TRIGONOMETRY
it is studied because it is very frequently
subjects in mathematics.
ANGLES
used in more advanced
2. Definitions.-The
definition of an angle as given in geometry
admits of a clear conception of small angles only. In trigonometry, we wish to consider positive and negative angles and these
Jf any size whatever; hence we need a more comprehensive
definition of an angle.
If a line, starting from the position OX (Fig. 1), is revolved
about the point 0 and always kept in the same plane, we say the
line generates an angle. If it revolves from the position OX to
the position OA, in the direction indicated by the arrow, the
angle XOA is generated.
YI
,A
The original position OX of the
generating line is called the initial
side, and the final position OA, the
x!
x terminal side of the angle.
If the rotation of the generating
line is counterclockwise, as already
C
taken, the angle is said to be positive.
If OX revolves in a clockwise direcy'
tion to a position, as OB, the angle
FIG. 1.
generated is said to be negative.
In reading an angle, the letter on the initial side is read first to
give the proper sense of direction.
If the angle is read in the
opposite sense, the negative of the angle is meant.
Thus,
LAOX = -LXOA.
It is easily seen that this conception of an angle makes it
possible to think of an angle as being of any size whatever.
Thus,
the generating line, when it has reached the position OY, having
made a quarter of a revolution in a counterclockwise direction,
has generated a right angle; when it has reached the position OX'
it has generated two right angles. A complete revolution generates an angle containing four right angles; two revolutions, eight
right angles; and so on for any amount of turning.
The right angle is divided into 90 equal parts called degrees (°),
each degree is divided into 60 equal parts called minutes ('), and
each minute into 60 equal parts called seconds (").
Starting from any position as initial side, it is evident that
for each position of the terminal side, there are two angles less
3
than 360°, one positive and one negative. Thus, in Fig. 1,
oe is the terminal side for the positive angle xoe or for the
negative angle xoe.
3. Quadrants.-It
is convenient to divide the plane formed by
a complete revolution of the generating line into four parts by
the two perpendicular lines X' X
Y
and Y' Y. These parts are
called first, second, third, and
II
Pz
fourth quadrants,
respectively.
They are placed as shown by the x~
x
Roman numerals in Fig. 2.
If OX is taken as the initial
IV
III
side of an angle, the angle is said
to lie in the quadrant in which its
fl
IY'
terminal side lies. Thus, XOP1
FIG. 2.
(Fig. 2) lies in the third quadrant,
and XOP2, formed by more than one revolution, lies in the first
quadrant.
An angle lies between two quadrants if its terminal side lies on
the line between two quadrants.
4. Graphical addition and subtraction of angles.-Two
angles
are added by placing them in the same plane
c
with their vertices together and the initialside
B
of the "econd on the terminal "ide of the first.
The sum is the angle from the initial side of
the first to the terminal side of the second.
.
0
Subtraction
is performed by adding the
0
A negative of the subtrahend to the minuend.
IL
FIG. 3.
Th
us, In
3,
'
' F Ig.
LAOB + LBOe = LAOe.
LAOe - LBOe = LAOe + LeOB = LAOE.
LBOe - LAOe = LBOe + LeOA = LBOA.
EXERCISES
Use the protractor
in laying off the angles in the L>llowing exercises:
1. Choose an initial side and layoff the following angles,
Indicate each
angle by a circular arrow.
75°; 145°; 243°; 729°; 456°; 976°. State the
quadrant in which each angle lies.
2. Layoff the following angles and state the quadrant that each is in:
-40°; -147°;
-295°;
-456°;
-1048°.
3. Layoff the following pairs of angles, using the same initial side for
each pair: 170° and -190°; -40° and 320°; 150° and -210°.
l
4
PLANE AND SPHERICAL TRIGONOMETRY
4.
Give a positive angle that has the same terminal side as each of the
following: 30°; 165°; -90°; -210°; -45°; 395°; -390°.
5. Show by a figure the position of the revolving line when it has generated each of the following: 3 right angles; 2i right angles; Ii right angles;
4i right angles.
Unite graphically, using the protractor:
6. 40° + 70°; 25° + 36°; 95° + 125°; 243° + 725°.
7. 75° - 43°; 125° - 59°; 23° - 49°; 743° - 542°; 90° - 270°.
8. 45° + 30° + 25°; 125° + 46° + 95°; 327° + 25° + 400°.
9. 45° - 56°
+
85°; 325°
-
256°
+
400°.
10. Draw two angles lying in the first quadrant
put differing by
360°.
Two negative angles in the fourth quadrant and differing by
360°.
11. Draw the following angles and their complements:
30°; 210°; 345°;
-45°; -300°;
-150°.
5. Angle measurement.-Several
systems for measuring angles
are in use. The system is chosen that is best adapted to the
purpose for which it is used.
(1) The right angle.-The
most familiar unit of measure of an
angle is the right angle. It is easy to construct, enters frequently
into the practical uses of life, and is almost always used in geometry. It has no subdivisions and does not lend itself readily to
computations.
(2) The sexagesimal system.-The
sexagesimal
system has
for its fundamental unit the degree, which is defined to be the
angle formed by -do part of a revolution of the generating linp
system used by eng;mecrs and others in making practical numerical computations.
The subdivisions of the degree
are the minute and the second, as stated in Art. 2. The word
"sexagesimal"
is derived from the Latin word sexagesimus,
meaning one-sixtieth.
(3) The centesimal system.-Another
system for measuring
angles was proposed in France somewhat over a century ago.
This is the centesimal system.
In it the right angle is divided
into 100 equal parts called grades, the grade into 100 equal parts
called minutes, and the minute into 100 equal parts called
seconds. While this system has many admirable features, its
use could not become general without recomputing with a
great expenditure of labor many of the existing tables.
(4) The circular or natural system.-In
the circular or natural
system for measuring angles, sometimes called radian measure or
.,,;-measure, the fundamental unit is the radian.
The radian is defined to be the angle which, when placed with
its vertex at the center of a circle, intercepts an arc equal in length
INTRODUCTION
5
to the radius of the circle. Or it itS defined as the positive angle
generated when a point on the generating line has passed through an
arc equal in length to the radius of the circle being formed by that
point.
In Fig. 4, the angles AOB, BOC, . . . FOG are each 1 radian,
since the sides of each angle intercept an arc equal in length to
the radius of the circle.
The circular system lends itself naturally to the measurement of angles in
many theoretical considerations.
It is
used almost exclusively in the calculus D
and its applications.
(5) Other systems.-Instead
of dividing the degree into minutes and seconds,
it is sometimes divided into tenths,
hundredths,
and thousandths.
This
decimal scale has been used more or less ever since decimal fractions were invented in the sixteenth century.
The mil is a unit of angle used in artillery practice.
The mil is
lr..roo revolution, or very nearly -rcrITOradian; hence its name.
The scales by means of which the guns in the United States Field
Artillery are aimed are graduated in this unit.
.
f"'T"I'I,
,'t
-
,-;YDLem
to UDein measuring an angle is apparent from a consideration of the geometrical basis for the definition of the radian.
FIG. 5.
FIG. 6.
(1) Given several concentric circles and an angle AOB at the
center as in Fig. 5, then
arc PlQl
OPl
=
arc P2Q2
OP2
=
arc PaQa
OPa
'
et c.
,
6
PLANE
AND
SPHERICAL
TRIGONOMETRY
INTRODUCTION
That is, the ratio of the intercepted arc to the radius of that arc
is a constant for all circles when the angle is the same. The angle
at the center which makes this ratio unity is then a convenient unit
for measuring angles. This is 1 radian.
(2) In the same or equal circles, two angl!3S at the center
are in the same ratio as their intercepted arcs. That is, in
Fig. 6,
LAOB
arc AB
LAOC - arc AC'
arc AB,
or,
r
in general, (J = ~, where (Jis the angle at the center measured in
r
radians, s the arc length, and r the radius of the circle.
7. Relations between radian and degree.-The
relations
between a degree and a radian can be readily determined from
their definitions.
Since the circumference of a circle is 211"times
the radius,
211"radians
Also
Then
360°
211"radians
.'.1
r
= 1 revolution.
= 1 revolution.
= 360°.
180°
=
0
11"
= 206264.8" + = 57° 17/ 44.8" +.
For less accurate work 1 radian is taken as 57.3°.
Conversely, 180°
= 7C radians.
. . . 1° =
1;0 = 0.0174533 - radian.
To convert radians to degrees, multiply the number of radians by
180
-, or 57.29578 -.
11"
To convert degrees to radians, multiply the number of degrees by
11"
an angle, it simply tells how many radians the angle contains.
Sometimes radian is abbreviated as follows: 3r, 3(r), 3p, or 3 rad.
When the word" radians" is omitted, the student should be careful to supply it mentally.
Many of the most frequently used angles are conveniently
expressed in radian measure by using
values are expressed accurately
Thus, 180° =
Here, if LAOC is unity when arc AC = r, LAOB
or 0.017453+.
180'
In writing an angle in degrees, minutes, and seconds, the signs
°, /, " are always expressed.
In writing an angle in circuhtr
measure, usually no abbreviation is used. Thus, the angle 2
means an angle of 2 radians, the angle
!11" means an angle of p
radians.
One should be careful to note that p does not denote
7
11"
11".
In this manner the
and long decimals are avoided.
radians, 90° =
!11"
radians, 60° =
111"
radians,
135° = tx- radians, 30° = t1l" radians.
These forms are more
convenient than the decimal form. For instance, 111" radians =
1.0472 radians.
Example I.-Reduce
2.5 radians to degrees, minutes, and
seconds.
Solution.-ll'adian
= 57.29578°.
Then 2.5 radians = 2.5 X 57.29578° = 143.2394°.
To find the number of minutes, multiply the decimal part of
the number of degrees by 60.
0.2394° = 60 X 0.2394 = 14.364/.
Likewise, 0.364/ = 60 X 0.364 = 21.8".
. . . 2.5 radians = 143° 14/ 22".
Example 2.-Reduce
22° 36/ 30" to radians.
Solut£on.-First.
change to degrees and decimal
This gives
22° 36' 30"
1°
22.6083° = 22.6083
. . . 22° 36/ 30"
=
=
X
=
of degree.
22.6083°.
0.017453 radian.
0.017453 = 0.3946 radian.
0.3946 radian.
EXERCISES
The first eight exercises are to be done orally.
1. Express the angles of the following numbers
iT;~;lr;tr;tr;~;¥r;tr.
of radians
2. Express the following angles as some number of
71"
in degrees:
radians: 30°; 90°;
180°; 135°; 120°; 240°; 270°; 330°; 225°; 315°; 81 °; 360°; 720°.
3. Express the angles of the following numbers of right angles in radians,
using 71";2; !; !; t; 3!; 21; Ii; 31.
4. Express in radians each angle of an equilateral triangle.
Of a regular
hexagon.
Of an isosceles triangle if the vertex angle is a right angle.
5. How many degrees does the minute hand of a watch turn through in
15 min.?
In 20 min.?
How many radians in each of these angles?
6. What is the measure of 90° when the right angle is taken as the unit
of measure?
Of 135°? Of 60°? Of 240°?
Of 540°? Of -270°?
Of
-360°?
Of -630°?
I
l
8
PLANE AND SPHERICAL TRIGONOMETRY
7. What is the measure of each of the angles of the previous exercise
when the radian is taken as the unit of measure?
8. What is the angular velocity of the second hand of a watch in radians
per minute?
What is the angular velocity of the minute hand?
Reduce the following angles to degrees, minutes and integral seconds:
9. 2.3 radians.
Ans. 131 ° 46' 49".
10. 1.42 radians.
Ans. 81° 21' 36".
11. 3.75 radians.
Ans. 214° 51' 33".
12. 0.25 radian.
Ans. 14° 19' 26".
13. T"~7I'radian.
Ans. 33° 45'.
14. -y..7I'radians.
Ans. 495°.
15. 0.0074 radian.
Ans. 25' 16".
16. 6.28 radians.
Ans. 359° 49' 3".
Reduce the following angles to radians correct to four decimals, using Art. 7 :
17. 55°.
18. 103°.
19. 265°.
20. 17°.
21. 24° 37' 27".
Ans. 0.4298.
22. 285° 28' 56".
Ans. 4.9825.
23. 416° 48' 45".
Ans. 7.2746.
Reduce the following angles to radians, using Table V, of Tables.
24. 25° 14' 23".
Ans. 0.4405162.
25. 175° 42' 15".
Ans. 3.0666162.
26. 78° 15' 30".
Ans. 1.3658655.
27. 243° 35' 42".
Ans. 4.2515348.
28. 69° 25' 8".
Ans. 1.2115882.
29. 9° 9' 9".
Ans. 0.1597412.
30. Compute the equivalents given in Art. 7.
31. Show that 1 mil is very nearly 0.001 radian, and find the per cent of
error in using 1 mil = 0.001 radian.
Ans. 1.86 per cent.
~-,~at
is thp_l11Pflsnrp of Pflrh of thp following "tlgl"s \\hcll the l'ight
---angle
is taken as the unit of measure: 1 radian, 271'radians, 650°, 2.157
radians?
Ans. 0.6366; 4; 7.222; 1.373.
33. An angular velocity of 10 revolutions per second is how many radians
per minute?
Ans. 3769.91.
34. An angular velocity of 30 revolutions
per minute is how many
11'
radians per second?
A ns. One-lf' radians.
35. An angular velocity of 80 radians per minute is how many degrees
per second?
Ans. 76.394°,
36. Show that nine-tenths the number of grades in an angle is the number
of degrees in that angle.
.
37. The angles of a triangle are in the ratio of 2: 3 ::7. Express the angles
in radians.
Ans. ilf'; tlf'; r721f'.
38. Express an interior angle of each of the following regular polygons in
radians: octagon, pentagon, 16-gon, 59-gon.
39. Express 48° 22' 25" in the centesimal system in grades, minutes, and
seconds.
Ans. 53 grades 74 min. 84 sec.
ANGLE
AT CENTER
OF CIRCLE
8. Relations between angle, arc, and radius.-In
Art. 6, it is
shown that, if the central angle is measured in radians and the arc
9
INTRODUCTION
length and the radius are measured in the same linear unit, then
arc
angle = -'-'
radlUs
That is, if 0, 8, and r are the measures, respectively,
arc, and radius (Fig. 7),
6 = s -;- r,
of the angle,
Solving this for 8 and then for r,
s = r6,
r = s -;- 6.
and
These are the simplest geometrical relations between the angle
at the center of a circle, the intercepted
arc, and the radius. They are of frequent use in mathematics and its applications, and should be remembered.
Example I.-The
diameter of a gradA
uated circle is 10 ft., and the graduations
are 5' of arc apart; find the length of arc
between the graduations in fractions of
an inch to three decimal places.
FIG. 7.
Solution.-By
formula, 8 = rO.
From the example, r = 12 X 5 = 60 in.,
and-
e = 0 01745~=
000145 rfJdifJn
Substituting in the formula, 8 = 60 X 0.00145 = 0.087.
. . . length of 5' arc is 0.087 in.
Example 2.-A train is traveling on a circular curve of !-mile
radius at the rate of 30 miles per hour.
B
Through what angle would the train turn
in 45 sec. ?
Solution.-When
at the position A (Fig.
Q 8), the train is moving in the direction AB.
After 45 sec. it has reached C, and is then
A moving in the direction CD. I t has then
r
turned through the angle BQC.
FIG.s.
Why?
But LBQC = LAOC = O.
The train travels the arc 8 = i mile in 45 sec.
To find value of 0, use formula
t
0
,
0=
(J
8 -;- r.
= -t + t = 0.75 radian = 42° 58' 19".
10
PLANE AND SPHERICAL TRIGONOMETRY
9. Area of circular sector.-In
Fig. 9, the area BOC, bounded
by two radii and an arc of a circle, is a sector. In geometry it is
shown that the area of a sector of a circle equals one-half the arc
length times the radius.
That
is,
But
Hence,
B
FIG. 9.
A
= !rs.
8 = reo
A = !r2e.
Example.-Find
the area of the sector
of a circle having a radius 8 ft. if the central
angle is 40°.
Solution.40° = 40 X 0.01745 = 0.698 radian.
Using the formula A = !r2e,
A = ! X 82 X 0.698
= 22.34.
. . . area of sector = 22.34 sq. ft.
ORAL EXERCISES
INTRODUCTION
2.
radius
A flywheel is revolving
of the wheel generate
How many
7r
11
at the rate of 456 r.p.m.
What angle does a
in 1 sec.?
Express in degrees and radians.
radians are generated in 2.5 sec.?
Ans. 2736°; 47.752 radians; 38.
3. A flywheel 6 ft. in diameter is revolving at an angular velocity of
30 radians per second.
Find the rim velocity in miles per hour.
Am. 61.36 miles per hour.
4. The angular velocity of a flywheel is 101r radians per second.
Find
the circumferential
velocity in feet per second if the radius of the wheel is
6 ft.
Am. 188.5 ft. per second.
5. A wheel is revolving
Find the number
at an angular
of revolutions
velocity
of 5; radians
per second.
per minute.
Per hour.
Ans. 50 r.p.m.; 3000 r.p.h.
6. In a circle of 9-in. radius, how long an arc will have an angle at the
center of 2.5 radians?
An angle of 155° 36'? Ans. 22.5 in.; 24.44 in.
7. An automobile wheel 2.5 ft. in outside diameter rolls along a road, the
axle moving at the rate of 45 miles per hour; find the angular velocity in
radians per second.
Ans. 16.81 7r radians.
7r
WRITTEN EXERCISES
8. Chicago is at north latitude 41
° 59'. Use 3960 miles as the radius
of the earth and find the distance from Chicago to the equator.
Ans. 2901.7 miles.
9. Use 3960 miles as the radius of the earth and find the length in feet
of I" of arc of the equator.
Ans. 101.37 ft.
10. A train of cars is running at the rate of 35 miles per hour on a curve of
1000 ft. radius.
Find its angular velocity in radians per minute.
Ans. 3.08 radians per minute.
11. Find tlw lpngth of arc which at 1 milp wiH Rl1nt('no an angl(' of I'.
An angle of I".
Ans. 1.536 ft.; 0.0253 ft.
12. The radius of the earth's orbit around the sun, which is about
92,700,000 miles, subtends at the star Sirius an angle of about 0.4".
Find
the approximate
distance of Sirius from the earth.
Ans. 48 (1012) miles.
13. Assume that the earth moves around the sun in a circle of 93,000,000mile radius.
Find its rate per second, using 365t days for a revolution.
Ans. 18.5 miles per second.
14. The earth revolves on its axis once in 24 hours.
Use 3960 miles for
the radius and find the velocity of a point on the equator in feet per second.
Find the angular velocity in radians per hour.
In seconds of angle per
second of time.
Ans. 1520.6 ft. per second; 0.262 radian per hour.
15. The circumferential
speed generally advised by makers of emery
wheels is 5500 ft. per minute.
Find the angular velocity in radians per
second for a wheel 16 in. in diameter.
Ans. 137.5 radian per second.
16. Find the area of a circular sector in a circle of 12 in. radius, if the
angle is 7r radians.
If 135°. If 5 radians.
1. The diameter of the drive wheels of a locomotive is 72 in. Find the
number of revolutions per minute they make when the engine is going 45
miles per hour.
Am. 210.08 r.p.m.
Am. 226.2 sq. in.; 169.7 sq. in.
17. The perimeter of a sector of a circle is equal to two-thirds the circumference of the circle.
Find the angle of the sector in circular measure and
in sexagesimal measure.
Am. 2.1888 radians; 125024.5'.
1. How many radians are there in the central angle intercepting
an arc
of 20 in. on a circle of 5-in. radius?
2. The minute hand of a clock is 4 in. long.
Find the distance moved
by the outer end when the hand has turned through 3 radians.
When it has
moved 20 min.
3. A wheel revolves with an angular velocity of 8 radians per second.
Find the linear velocity of a point on the circumference if the radil1R iR f\ ft.
4. The veloeity of the rim of a flywheel is 75 ft. per second.
Find the
angular velocity in radians per second if the wheel is 8 ft. in diameter.
5. A pulley carrying a belt is revolving with an angular velocity of 10
radians per second.
Find the velocity of the belt if the pulley is 5 ft. in
diameter.
6. An angle of 3 mils will intercept what length of arc at 1000 yd.?
7. A freight car 30 ft. in length at right angles to the line of sight intercepts an angle of 2 mils.
What is its distance from the observer?
8. A train is traveling on a circular curve of !-mile radius at the rate of 30
miles an hour.
Through what angle does it turn in 15 sec.?
9. A belt traveling 60 ft. per second runs on a pulley 3 ft. in diameter.
What is the angular velocity of the pulley in radians per second?
10. A circular target at 3000 yd. sub tends an angle of 1 mil at the eye.
How large is the target?
~
12
PLANE
AND SPHERICAL TRIGONOMETRY
10. General angles.-In
Fig. 10, the angle XOP1 is 30°; or if
the angle is thought of as formed by one complete revolution and
30°, it is 390°; if by two complete revolutions and 30°, it is 750°.
So an angle having OX for initial side and OP1 for terminal side
+ 30°,
is 30°,360°
2 X 360°
+ 30°,
or, in general,
n X 360° + 30°,
where n takes the values 0, 1, 2, 3, . . . , that is, n is any integer,
zero included.
In radian
measure
this is 2n1l"
+
t1l".
The expression n X 360° + 30°, or 2n1l" + t1l", is called the
general measure of all the angles ha ving OX as initial side and
.
OP1 as terminal side.
If the angle XOP2 is 30° less than 180°, then the general measure of the angles having OX as initial side and OP2 as terminal
y
side is an odd number times 180°
less 30°; and may be written
~
(2n + 1)180° - 30°,
X
an
integral number of times 11" is taken
and then t1l" is added or subtracted.
This gives the terminal side in one
y'
of the four positions shown in Fig.
FIG. 10.
~~_by OP1, OP2, OP~, and OPI.
it is evident that throughout this article n may have negative
as well as positive values, and that any angle () might be used
instead of 30°, or t1l".
EXERCISES
1. Use the same initial side for each and draw angles of 50°; 360° + 50°;
n . 360° + 50°,
2. Use the same initial side for each and draw angles of 40°; 180° + 40°;
2. 180° + 40°; 3 . 180° + 40°; n . 180° + 40°.
3. Use the same initial side for each and draw angles of 30°; 90° + 30°;
2, 90° + 30°; 3 .
90° + 30°; n . 90° + 30°.
4. Draw the terminal sides for all the angles whose general measure is
(2n
+
(4n
-
1)7r :!: 1
3"';
7r
1)2 :!:
7r
6'
2n7r
1
+ 3"';
(2n
+
1
+
+
IH...
1)180° :!: 60°;
1)2 :!: 3; n7r:!: 4""; (4n
"'7r
+
11. Directed lines and segments.- For certain purposes in
trigonometry it is convenient to give a line a property not often
used in plane geometry.
This is the property of having direction.
In Fig. 11, RQ is a directed straight line if it is thought of as
traced by a point moving without change of direction from R
toward Q or from Q toward R. The direction is often shown by
an arrow.
Let a fixed point 0 on RQ be taken as a point from which to
measure distances.
Choose a fixed length as a unit and lay it
off on the line RQ beginning at O. The successive points located
in this manner will be 1, 2, 3, 4, . . . times the unit distance
0
~
I) Q
R P Pz
-5
R;3
(4n + 3H....
6. Draw the following angles: 2n X 180° :!: 60°; (2n
COORDINATES
F':J
(2n + 1)11"- t1l".
Similarly,
n1l" ::t t1l" means
2n . 90°. For all the angles whose general measure is (2n + 1)90°.
5. Draw the following angles: 2n...; (2n + 1)...; (2n
+ In...; (4n
7. Give the general measure of all the angles having the lines that bisect
the four quadrants as terminal sides.
Those that have the lines that trisect
the four quadrants as terminal sides.
I
or
x'
13
INTRODUCTION
1)2"'7r :!: 6;
I
-4
I
I
I
I
I
I
I
I
-3
-2
-1
0
1
2
3
4
.
FIG. 11.
from O. These points may be thought of as representing the
numbers, or the numbers may be thought of as representing
the points.
Since there are two directions from 0 in which the measurements m2j' be m~de, it is e,'ioent thRt there are two points cflually
distant from O. Since there are both positive and negative numbers, we shall agree to represent the points to the right of 0 by
positive numbers and those to the left by negative numbers.
Thus, a point 2 units to the right of 0 represents the number 2;
and, conversely, the number 2 represents a point 2 units to the
right of O. A point 4 units to the left of 0 represents the number
-4; and, conversely, the number -4 represents a point 4 units
to the left of O.
The point 0 from which the measurements are made is called
the origin. It represents the number zero.
A segment of a line is a definite part of a directed line.
The segment of a line is read by giving its initial point and
its terminal point. Thus, in Fig. 11, OP1, OP2, and P\Pa are
segments.
In the last, PI is the initial point and Pa the terminal
point.
The value of a segment is determined by its length and direction, and it is defined to be the number which would represent the
terminal point of the segment if the initial point were taken as origin.
. I
I I
I
l
14
PLANE
AND SPHERICAL TRIGONOMETRY
INTRODUCTION
It follows from this definition that the value of a segment read
in one direction is the negative of the value if read in the opposite
direction.
In Fig. 12, taking 0 as origin, the values of the segments are as
follows:
OPI = 3, OP3 = 8, OP5 = -5, P?3 = 3, P~l = -5.
P~6 = -6, PsP5 = 3, PIP2 = -P?l = 2.
Two segments
the same length,
If two segments
is on the terminal
are equal if they have the same direction and
that is, the same value.
are so placed that the initial point of the second
point of the first, the sum of the two segments
. . P,;, . . Ps. . , ~o, , , . , ~~, . , . . Pa, , .
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 I 2 3 4 5 6 7 8 9 10
FIG. 12.
is the segment having as initial point the initial point of the first,
and as terminal point the terminal point of the second.
The segments are subtracted by reversing the direction of the
subtrahend and adding.
Thus, in Fig. 12,
.
PsP4
+ P~l = PsPl
+
= 8.
= P?6 = -13.
PIP;>- P,j>. = PtP3 + PJ>2
= PIP2 = 2.
P?3 - PIP3 = P?3 + P~l = P2PI = -2.
12. Rectangular coordinates.-Let
X' X and Y' Y (Fig. 13)
be two fixed directed straight lines, perpendicular to each other
and intersecting at the point O. Choose the positive direction
towards the right, when parallel to X' X; and upwards, when
parallel to Y'Y.
Hence the negative directions are towards the
left, and downwards.
The two lines X'X and Y'Y divide the plane into four quadrants, numbered as in Art. 3.
Any point PI in the plane is located by the segments NPI and
MPI drawn parallel to X'X and Y'Y respectively, for the values
of these segments tell how far and in what direction PI is from the
two lines X'X and Y'Y.
It is evident that for any point in the plane there is one pair
of values and only one; and, conversely, for every pair of values
there is one point and only one.
P2P4
15
The value of the segment NPI or OM is called the abscissa of
the point PI, and is usually represented by x. The value of the
segment MPI or ON is called
y
the ordinate of the point PI,
and is usually represented
~.
by y. Taken together the
~
N
abscissa x and the ordinate y
are called the coordinates of
Q
x
the point Pl. They are writ- x'
0
M
ten, for brevity, within parentheses and separated by a
Pa'
comma, the abscissa always
'~
being first, as (x, y).
IY'
The line X' X is called the
FIG. 13.
axis of abscissas or the x-axis.
The line Y' Y is called the axis of ordinates or the y-axis.
Together, these lines are called the coordinate axes.
It is evident that, in the first quadrant, both coordinates are
positive; in the second quadrant, the abscissa is negative and
the ordinate is positive; in the third quadrant, both coordinates
are negative; and, in the fourth quadrant, the abscissa is positive
and the ordinate is negative. This is shown in the following table:
P4P6
r
I
~uadrant
I
I
I
II
I III I IV
-
Abscissa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ordinate
.. .
/
t ~
I
+
=
I
I
Thus, in Fig. 13, PI, P2, P3, andP4 are, respectively, the points
y
(4, 3), (-2, 4), (-4,
-3), and
(3, -4).
The points M, 0, N, and
~
Q are, respectively, (4, 0), (0, 0),
(0, 3), and ( -4, 0).
y
13. Polar coordinates.-The
x~
M-X
point PI (Fig. 14) can also be
0
located if the angle (Jand the length
of the line OPI are known. The
line OPt is called the radius vector
y'
and is usually represented by r.
FIG.14.
Since r denotes the distance of
the point PI from 0, it is always considered positive.
. I
I I
I
l
14
PLANE
AND SPHERICAL TRIGONOMETRY
INTRODUCTION
It follows from this definition that the value of a segment read
in one direction is the negative of the value if read in the opposite
direction.
In Fig. 12, taking 0 as origin, the values of the segments are as
follows:
OPI = 3, OP3 = 8, OP5 = -5, P?3 = 3, P~l = -5.
P~6 = -6, PsP5 = 3, PIP2 = -P?l = 2.
Two segments
the same length,
If two segments
is on the terminal
are equal if they have the same direction and
that is, the same value.
are so placed that the initial point of the second
point of the first, the sum of the two segments
. . P,;, . . Ps. . , ~o, , , . , ~~, . , . . Pa, , .
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 I 2 3 4 5 6 7 8 9 10
FIG. 12.
is the segment having as initial point the initial point of the first,
and as terminal point the terminal point of the second.
The segments are subtracted by reversing the direction of the
subtrahend and adding.
Thus, in Fig. 12,
.
PsP4
+ P~l = PsPl
+
= 8.
= P?6 = -13.
PIP;>- P,j>. = PtP3 + PJ>2
= PIP2 = 2.
P?3 - PIP3 = P?3 + P~l = P2PI = -2.
12. Rectangular coordinates.-Let
X' X and Y' Y (Fig. 13)
be two fixed directed straight lines, perpendicular to each other
and intersecting at the point O. Choose the positive direction
towards the right, when parallel to X' X; and upwards, when
parallel to Y'Y.
Hence the negative directions are towards the
left, and downwards.
The two lines X'X and Y'Y divide the plane into four quadrants, numbered as in Art. 3.
Any point PI in the plane is located by the segments NPI and
MPI drawn parallel to X'X and Y'Y respectively, for the values
of these segments tell how far and in what direction PI is from the
two lines X'X and Y'Y.
It is evident that for any point in the plane there is one pair
of values and only one; and, conversely, for every pair of values
there is one point and only one.
P2P4
15
The value of the segment NPI or OM is called the abscissa of
the point PI, and is usually represented by x. The value of the
segment MPI or ON is called
y
the ordinate of the point PI,
and is usually represented
~.
by y. Taken together the
~
N
abscissa x and the ordinate y
are called the coordinates of
Q
x
the point Pl. They are writ- x'
0
M
ten, for brevity, within parentheses and separated by a
Pa'
comma, the abscissa always
'~
being first, as (x, y).
IY'
The line X' X is called the
FIG. 13.
axis of abscissas or the x-axis.
The line Y' Y is called the axis of ordinates or the y-axis.
Together, these lines are called the coordinate axes.
It is evident that, in the first quadrant, both coordinates are
positive; in the second quadrant, the abscissa is negative and
the ordinate is positive; in the third quadrant, both coordinates
are negative; and, in the fourth quadrant, the abscissa is positive
and the ordinate is negative. This is shown in the following table:
P4P6
r
I
~uadrant
I
I
I
II
I III I IV
-
Abscissa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ordinate
.. .
/
t ~
I
+
=
I
I
Thus, in Fig. 13, PI, P2, P3, andP4 are, respectively, the points
y
(4, 3), (-2, 4), (-4,
-3), and
(3, -4).
The points M, 0, N, and
~
Q are, respectively, (4, 0), (0, 0),
(0, 3), and ( -4, 0).
y
13. Polar coordinates.-The
x~
M-X
point PI (Fig. 14) can also be
0
located if the angle (Jand the length
of the line OPI are known. The
line OPt is called the radius vector
y'
and is usually represented by r.
FIG.14.
Since r denotes the distance of
the point PI from 0, it is always considered positive.
II
16
PLANE
AND
SPHERICAL
TRIGONOMETRY
Point 0 is called the pole. The corresponding values of rand
(Jtaken together are called the polar coordinates of the point P.
It is seen that r is the hypotenuse of a right triangle of which
x and yare the legs; hence r2 = x2 + y2, no matter in what
quadrant
the point
is located.
EXERCISES
1. Plot the points (4, 5), (2, 7), (0, 4), (5, 5), (7, 0), (-2, 4), (-4, 5),
(-6, -2), (0, -7), (-6, 0), (3, -4), (7, -6).
2. Find the radius vector for each of the points in Exercise 1. Plot
in each case.
Ans. 6.40; 7.28; 4; 7.07.
3. Where are all the points whose abscissas are 5? Whose ordinates .are
O?
Whose abscissas are -2?
Whose radius vectors are 3?
4. The positive direction of the x-axis is taken as the initial side of an
angle of 60°. A point is taken on the terminal side with a radius vector
equal to 12. Find the ordinate and the abscissa of the point.
6. In Exercise 4, what is the ratio of the ordinate to the abscissa?
The
ratio of the radius vector to the ordinate?
Show that you get the same
ratios if any other point on the terminal side is taken.
6. With the positive x-axis as initial side, construct angles of 30°, 135°,
240°, 300°. Take a point on the terminal side so that the radius vector is
2a in each case, and find the length of the ordinate and the abscissa of the
point.
7. The hour hand of a clock is 2 ft. long.
Find the coordinates
of its
outer end when it is twelve o'clock; when three; nine; half-past ten.
Use
perpendicular
and horizontal axes intersecting where the hands are fastened
Ans. (0.2); (2, 0); (-2,0);
(-1.414,
1.414).
CHAPTER
TRIGONOMETRIC
II
FUNCTIONS
OF ONE ANGLE
14. Functions of an angle.-Connected
with any angle there
are six ratios that are of fundamental importance, as upon them
is founded the whole subject of trigonometry.
They are called
trigonometric ratios or trigonometric functions of the angle.
One of the first things to be done in trigonometry is to investigate the properties of these ratios, and to establish relations
y
y
M
0
x
x
(b)
(a)
y
y
8
/~
M
x
/
/'
l
~
"',
~""\
MX
(d)
(c)
FIG. 15.
between them, as they are the tools by which we work all sorts
of problems in trigonometry.
15. Trigonometric ratios.- Draw an angle 8 in each of the
four quadrants as shown in Fig. 15, each angle having its vertex
at the origin alld its initial side coinciding with the positive part
of the x-axis. Choose any point P (x, y) in the terminal side of
such angle at the distance r from the origin. Draw MP ..L OX,
forming the coordinates
OM = x and MP = y, and the radius
vector, or distance, OP = r. Then in whatever quadrant (J is
found, the functions are defined as follows:
17
l
'il
18
PLANE
.
sme
AND SPHERICAL TRIGONOMETRY
.
6 ( wntten
.
sm
6)
= ordinate
MP
y
ISta nce = OP =-0r
abscissa
OM
x
d'
'.
cosme
6 (wntten
cos 6)
=
d' IS t ance
.
6 (wntten
= OP =-.l'
ordinate
MP
y
tangent
tan 6)
=
.
=
=
-.
abscIssa
OM
x
.
abscissa
OM
x
cotangent 6 (wntten cot 6)
= or d' mate =
MP = y-.
.
distance
OP
r
secant 6 (wntten sec 6)
= abscissa = -OM =-.x
.
distance
OP
r
cosecant 6 (wntten
csc 6)
= or mate
=
-.
d'
= MP y
Two other functions frequently used are:
versed
sine 6 (written
vers 6)
coversed sine 6 (written covers 6) = 1 - cos 6.
= 1 - sin 6.
The trigonometric functions are pure numbers, that is, abstract
numbers, and are subject to the ordinary rules of algebra, such
Pyas1j
addition, subtraction, multi plicaJ
tion, and division.
16. Correspondence
bet wee n
angles and trigonometric
ratios.
X To each and every angle there corre8p(}nrf.~
but
()
rzgonometrzc ratio. Draw any angle
FIG.16.
(J as in Fig. 16. Choose points
PI, P2, P3, etc. on the terminal side OP. Draw MIPI, M ?2,
M ?3, etc. perpendicular to OX. From the geometry ofthefigure,
TRIGONOMETRIC FUNCTIONS OF ONE ANGLE
be handled by methods of geometry.
Geometry gives but
few relations between angles and lines that can be used in computations, as most of these relations are stated in a comparative
manner-for
instance, in a triangle, the greater side is opposite
the greater angle.
Definition.-When
one quantity so depends on another that
for every value of the first there are one or more values of the
second, the second is said to be a function of the first.
Since to every value of the angle there corresponds a value for
each of the trigonometric ratios, the ratios are called trigonometric functions.
They are also called natural trigonometric functions in order to
distinguish them from logarithmic trigonometric functions.
A table of natural trigonometric functions for angles 0 to 90°
for each minute is given on pages 112 to 134 of Tables. * An
explanation of the table is given on page 29 of Tables.
17. Signs of the trigonometric functions.-The
sine of an
angle (Jhas been defined as the ratio of the ordinate to the distance of any point in the terminal side of the angle. Since the
distance r is always positive (Art. 13), sin (Jwill have the same
algebraic sign as the ordinate of the point. Therefore, sin () is
positive when the angle is in the first or second quadrant, and
negative when the angle is in the third or fourth quadrant.
''-
tions of () are determined.
lowing table:
OM I = OM2 = OM3 = etc. = tan (J,
and similarly for the other trigonometric ratios. Hence, the six
ratios remain unchanged as long as the value of the angle is
unchanged.
It is this exactness of relations between angles and certain lines
connected with them that makes it possible to consider a great
variety of questions by means of trigonometry which cannot
. .. . . . . . . . . . . . . . . . . . . ..
II.....
.. .. .. .. .. .... .. . . ..
III.... .. .. .... .. .. .. .... ..
IV.
""""""""""'"
~
should verify the fol-
cos (J tan (J cot (J sec (J
sin (J
I
I...
''-,'t..
~
The student
Quadrant
MIPI
M?2
M?3
.
OPI = OP2 = OP3 = etc. = sm (J,
OM I
OM2
OM3
OPI = OP2 = OP3 = etc. = cos (J,
MIPI
M?2
M3P3
19
I
I
I
I
+
+
+
+
+
-
-
-
+
+
+
-
+
+
-
-
csc
(J
I
+
+
-
-
It is very important that one should be able to tell immediately
the sign of any trigonometric function in any quadrant.
The
signs may be remembered by memorizing the table given; but,
for most students, they may be more readily remembered by
discerning relations between the signs of the functions.
One
* The reference
authors.
is to "Logarithmic
and Trigonometric
Tables"
by the
II
20
PLANE
AND
SPHERICAL
TRIGONOMETRY
TRIGONOMETRIC
good scheme is to fix in mind the signs of the sine and cosine.
Then if the sine and cosine have like signs, the tangent is plus;
and if they have unlike signs, the tangent is negative.
The
signs of the cosecant, secant, and cotangent always agree respectively with the sine, cosine, and
y
tangent.
The scheme shown in
Fig. 17 may help in remembersm +
sin +
ing the signs.
tan tan +
}
C08+ }
cos -
cos- }
sin-
tan +
cos+ }
tan-
FIG. 17.
3. When
4. When
6. When
6. When
7. When
Give the
angles:
8. 120°.
sin
cos
sin
sin
see
sign
() is positive and
() is positive and
() is negative and
() is negative and
() is negative and
of each of the
12. i....
:J
16. \t....
20. 2n... +~....
10. 340°.
14. J~l,r.
18. -213°.
22. (2n + 1)... - i....
11. 520°.
16. t
19. -700°.
23. (2n + 1)... + i....
24. Show that neither the sine nor the cosine of an angle can be greater
than + 1 or less than -1.
26. Show that neither the secant nor the cosecant of an angle
value between -1 and + 1.
26. Show that the tangent and the cotangent
of an angle
any real value whatever.
27. Is there an angle whose tangent is positive and whose
is negative?
Whose secant is positive and whose cosine is
Whose secant is positive and whose cosecant is negative?
Construct and measure the following acute angles:
28. Whose sine is i.
31. Whose cotangent is 3.
29. Whose tangent is f.
32. Whose secant is t.
30. Whose cosine is j.
33. Whose cosecant is i.
can have a
may
have
cotangent
negative?
COMPUTATIONS OF TRIGONOMETRIC FUNCTIONS
18. Calculation from measurements.
Example.-Determine
the approximate values of the functions of 25°. By means of the
protractor draw angle KOP = 25° (Fig. 18). Choose P in the
side, say, 2-1\ in. distant
MP 1. OX.
By measurement,
OM
21
OF ONE ANGLE
from
=
the origin.
2 in. and
MP
Draw
=H
in.
From the definitions we have:
H
00M
2
= MP
2
0 91
OP = 2T\ = 0.43. cos 5 = OP = 2-1\ = . .
.
sm 25 °
tan 25° =
MP
H
7
OM = 2" = 0.4.
sec 25° =
g~ = zr
x
0
.in-
EXERCISES
Answer Exercises 1 to 27 orally.
In what quadrant does the terminal
side of the angle lie in ench of the
following cases:
1. When all the functions
are
positive?
2. When sin () is positive and cos ()
negative?
tan () negative?
tan () negative?
tan () positive?
cos () negative?
csc () negative?
trigonometric
functions of the following
terminal
FUNCTIONS
00M
2
cot 25 = MP = H
= 1.09. csc 25° =
nl"
;; ~
=
= 2 .13 .
= 2.33.
vers 25° = 1 - cos 25° = 1 - 0.91 = 0.09.
covers 25° = 1 - sin 25° = 1 - 0.43 = 0.57.
In a similar manner any angle can be constructed, measurements taken, and the functions computed; but the results will be
~nly approximate
maccuracy
because of the
of measurement.
I
Y
EXERCISE
In the same figure construct angles of 10°,
20°, 30°, . . . 80°, with their vertices at the
origin and their initial sides on the positive
part of the x-axis.
Choose the same distance
FIG. 18.
on the terminal side of each angle, draw and
measure the coordinates,
and calculate the trigonometric
functions
of each
IV.
19. Calculations from geometric relations.-There
are two
right triangles for which geometry gives definite relations between
"ides and angles. These are the right isosceles triangle whose
acute angles are each 45°, and the right triangles whose acute
angles are 30 and 60°. The functions of any angle for which the
abscissa, ordinate, and distance form one of these triangles can
readily be computed to any desired degree of accuracy.
All
such angles, together with 0, 90, 180, 270, and 360°, with their
functions are tabulated on page 24. These are very important
for future use.
20. Trigonometric functions of 300.-Draw
angle XOP = 30°
as in Fig. 19. ChooseP in the terminal side and draw MP 1. OX.
By geometry, MP, the side opposite the 30°-angle, is one-half the
hypotenuse OP. Take y = MP = 1 unit. Then r = OP = 2
units, and x = OM = 0.
By definition. then, we have:
l
22
sin 30°
PLANE AND SPHERICAL TRIOONOMETRY
-1
= 1J.
r - 2'
cot 30°
=
~
0
=
TRIGONOMETRIC
y3
tan 120° = 1J.=
x
-1
\3 = 0.
1
see 30° = ~ = ~ = ~0 .
x 0
3
~
0
1
tan 30° = Jf =
.
= ~
x
0 -- 3 = 30. csc 30° = !..
Y 1= 2
21. Trigonometric functions of 45°.-Draw angle XOP = 45°
as in Fig. 20. Choose the point P in the terminal side and draw
its coordinates OM and MP, which are necessarily equal. Then
cos 30° =
::. -
r
y
p
y
,300
0
va
M
X
MX
FIG. 19.
the coordinates
FIG. 20.
of P may be taken as (1, 1), and r = y2.
definition, then, we have:
sin 45° = -Y = - 1
r
0
cos 45° =
::.
r
cot 120° =
= 20.
2
1
= 20.
cot 45° =
1
=~
V2 = 20.
:£
Y
=
sec 45° = !.. =
T
Y
1
tan 45° = ;;=1=1.
!
1
By
= 1.
v0 = 0.
1
0 = v.- 1 -2
Y
1
22. Trigonometricfunctions
of 120°.-Draw
angle XOP
= 120°
as
in
Fig.
21.
Choose
any
point P in the
y
terminal side and draw its coordinates OM
and MP. Triangle MOP is a right triangle
)200 with LMOP = 60°. Then, as in computing
...
X the functions of 30°, we may take OP = 2,
MO = 1, and MP = 0.
But the abscissa
of P is OM = -1.
Then the coordinates
FIG. 21.
of Pare (-1, 0),
and r = 2. By definition, then, we have:
sin 120° =
csc 45° = ~ =
V; = ~,,13.
~ =
x
-1
cos 120 ° = - = r
2
1
= --.
2
FUNCTIONS
::.
Y
=
OF ONE ANGLE
= --v
-1
23
jCj
0).
3
= -!0.
0
3
sec 120° = ~ = 2 = - 2 .
x
-1
csc 120° = !.. =
Y
~
0
.
= ~0
3
In forming the ratios for the angles whose terminal sides lie
on the lines between the quadrants, such as 0, 90, 180, 270, and
360°, the denominator is frequently zero.
y
Strictly speaking, this gives rise to an
impossibility
for division by zero is
meaningless.
In all such cases we say
P(a,o)
that the function has become infinite.
00
0
23. Trigonometric
functions of 0°.The initial and terminal sides of 0° are
both on OX. Choose the point P on OX
FIG. 22.
as in Fig. 22, at the distance of a from O.
Then the coordinates of P are (a, 0), and r = a. By definition,
then, we have:
0
~ = O.
tan {}O= Y
:;: = ;; = O.
r
a
x
a
ora
cos 0 ° = - = - = 1.
see 0 = - = - = 1.
r
a
x
a
cot 0° and csc 0° have no meaning.
*
24. Trigonometric functions of 90°.-Draw
angle XOY = 90°
as in Fig. 23. Choose any point P in the terminal side at
sin {}O = 1J. =
*
By the expression
-
00 is understood
~ =
;
as x approaches
I = a; 0~1 = lOa; 0.~1 = 100a; 0.;01
zero as a limit.
For example,
1000a;
= 10,000,OOOa; etc.
0.00;0001
the value of
=
That is, as x gets nearer and nearer
to zero ~ gets larger and larger, and can be made to become larger than any
x
number
zero.
N.
The value of ~ is then said to become infinite
x
The symbol is
00
usually read infinity.
as x approaches
It should be carefully noted
that a is not divided by 0, for division by 0 is meaningless.
Whenever the symbol"
00
is used it should be read "has no meanin~."
"
l
24
PLANE AND SPHERICAL
FREQUENTLY
0 in
radians
0°
0°
0
30°
7r
4
60°
7r
90°
1
2
0
2
0
2
2",.
120°
"3
3",.
136°
4
160°
fur
6"
180°
7r
7",.
210°
6"
5",.
226°
4
4",.
240°
"3
37r
"2
5",.
300°
330°
va
T
:3
"3
7",.
4
II",.
6
360°
2",.
1
2
0
1
-2
0
-2
0
-2
1
-2
0
-2
0
-2
-1
0
-2
0
-2
1
-2
-1
1
-2
0
1
2
cot 0
0
sec 0
0
1
2
0
V2
a-
2
0
00
00
0
00
20
-a-
0
va
20
-a-
-0
-a-
-2
20
-r
-1
-1
-0
0
-0
20
--a-1
20
0
-a0
0
a-
00
0
1
1
0
0
"3"
0
00
0
-0
-a-
0
2
0
2
-1
-1
0
-a-
-0
1
0
00
=~I
-2
00
2
0
20
~
1
""f
FIG. 23.
x
sin 90°
=
J!..- a
r-(i=I.
0
cot 90° = y:: -- (i
=aO.
rx = (i0 = O.
25
2
00
-2
-0
20
--a-1
_20
_;1
-:J
x
r
a
EXERCISES
Construct the figure and compute the functions for each of the following
angles.
1. 60°
2. 135°.
3. 150°.
4. 180°.
6. 240°.
6. 330°.
7. 270°.
8. 315°.
25. Exponents of trigonometric functions.- When the trigonometric functions are to be raised to powers, they are written
sin2 8, cos3 8, tan4 8, etc., instead of (sin 8)2, (cos 8)3, (tan 8)4,
etc., except when the exponent is -1.
Then the function is
enclosed in parentheses.
the distance a from the origin. Then the coordinates of Pare
YI
(0, a), and r
P(o,a)
= a. By definition, then,
we have:
0
OF ONE ANGLE
csc 90 ° --=-- y
a-' 1
tan 90° and sec 90° have no meaning.
csc 0
1
00
3
FUNCTIONS
FUNCTIONS
cos 90° --
0
va
0
0
-2
0
-2
AND THEIR
tan
1
v'3
T
V2
T
1
2
0
T
1r
ANGLES
cas 0
0
1
2
V2
6
46°
316°
sin 0
7r
TRIGONOMETRY
TRIGONOMETRIC
USED
Thus, (sin 8)-1
= SIn
~
8
(see Art. 35).
EXERCISES
Find that the numerical values of each of the Exercises 1 to 10 is unity.
1. sin2 3(}° + cos2 30°.
6. sec230°
- tan2 30°.
2. sin2 60° + COS2 60°.
3. sin2 120° + cos2 120°.
4. sin2 135° + cos2135°.
6. sin2 300° + COS2 300°.
Find
oppimRl
the
numerical
values
7. sec2 150° - tan2 150°.
8. sec2330° - tan2 330°.
9. csc245° - cot2 45°.
10. csc2 240°
of the
following
- cot2 240°.
expressions
correct
11. sin 45° + 3 cas 60°.
12. cos2 60° + sin3 90°.
Ans. 2.207.
Ans. 1.250.
13. 10 cas' 30° + see 45°.
Ans.
Ans.
14. see 0° . cas 60° + csc 90° sec2 45°.
16. cas 120° cas 270° - sin 90° tan3 135°.
Ans.
In the following expressions, show that the left-hand member
to the right, by using the table on page 24:
16. sin 60° cas 30°
+
cas 60° sin 30°
17.
18.
19.
20.
= sin
cas 45° cas 135° - sin 45° sin 135°
sin 60° cas 30° - cas 60° sin 30° =
cas 210° cas 30° - sin 210° sin 30°
sin 300° cas 30° - cas 300° sin 30°
21. tan 240° + tan 60°
1 - tan 240° tan 60° = t an 300 ° .
tan 120° - tan 60°
22.
1 + tan 120° tan 60° = tan 60°.
26. Given the function
angle.
to three
plRN'~:
Example I.-Given
find the other functions.
7.039.
2.500.
1.000.
is equal
90°.
= cas 180°.
sin 30°.
= cas 240°.
= sin 270°.
of an acute angle, to construct the
sin 8 = t. Construct angle () and
~
26
PLANE
AND SPHERICAL
TRIGONOMETRY
TRIGONOMETRIC FUNCTIONS
i. Since weare
of the lines, we may take y
Solution.-By
sin ()
definition,
=
~ =
concerned
,
LXOP
'.
() is the required
=
N
P,
,
B
SIn ()
3
x
M
Example
FIG. 24.
Y OP2 -
= Y25 - 16 = 3.
angle since sin ()
=
The remaining functions may be written as follows:
()
3.-Given
tan ()
~: = ~.
SIn
V2§'
cos tJ = -~,
V2§
X
OM
3
MP
4
OM
3
= -OP = -,5 tan () = OM
- = 3-, cot () = MP
-,
4
- =
OP
5
OP
5
()
sec () = -,
csc
= -MP = 4-.
OM = 3
Example 2.-Given
cas ()
= j. Construct angle () and find
the other functions.
()
i.
Choose x = 2 and
OY and 2 units to the right (Fig. 25),
intersecting OX at M. With the origin as a center and a radius
of 3 units, draw an arc cutting AB at P. Join 0 and P, forming
LXOP.
Then OP = 3, OM
II
= 2, and
csc
,)
FIG. 25.
r = 3. Draw AB
i.
() and find the
angle
YI
A
p
~
()
D
J)
M X
!
B
0
The other functions are as follows:
1M
B
=~ =
Construct
,',
V5
cas ()
= t.
C
OP = y'29.
LXOP = (J is the required angle
MP
~.
since tan () =
OM = 5
p
definition,
=
OP
3
OP
3
= -OM = -,2 csc () = -MP = -.
Y5
II
A
Y
Solution.-By
angle since cos () = ~Af
intersecting OX at M; also draw CD
OX and 2 units above intersecting AB
at P. Then OM = 5, MP = 2, and
Mpi
. '. LXOP = () is the required
cos
= Y5.
other functions.
Solution.-By definition, tan (J = =~. Choose y = 2 and
~
x = 5. Draw AB" OY and 5 units to the right (Fig. 26),
a radius of 5 units, draw an arc intersecting AB in the point P.
Draw OP forming LXOP, and draw MP J.. OX. Then OP
MP = 4, and
= 5,
OM =
27
MP
y5
MP
V5
OM
2
= -OP = -,3 tan () = -OM = -, 2 cot () = MP
= -,
V5
sec
0
OM2
ANGLE
The remaining functions are as follows:
,Y
A
-
MP = YOp2
only with the ratios
and above
r = 5
units of any size. Draw AB parallel to OX and= 44,units
(Fig. 24), intersecting OY at N. With the origin as a center and
OF ONE
FIG. 26
cot (J
= -,
2
sec (J =
g-'
V2§
= -. 2
EXERCISES
In Exercises 1 to 12, construct ()from the given function and find the other
functions of () when in the first quadrant,
1. sin () = i.
9. sec () = 0,
5. sin () = !'
2. cos () = i,
cos () = h/a.
10. csc () = ~,
()
a
()
3. tan
3.
7. tan ()
6.
=
.
11. tan
= Ii'
4. cot () = 2,5.
a
8. cos ()
= b'
sin()cos()
of
, when
sec () csc ()
~
= 4.
12. cot () = I. .
1.
() IS an acute
= 5 and
angle.
Ans. 11.'
.
sec()+tan()
3.
()
_
14. Fmd the value of
()
IS an acute
= 5 and
cos () + vers ()' when cos
'
angle.
Ans.3.
13. Fmd
the
value
tan
()
_'
II
28
PLANE
AND
TRIGONOMETRY
SPHERICAL
TRIGONOMETRIC
.
15. Fmd the v/I,)ue of
csc IJ
+
see IJ
sin IJ + cas fi' when cas IJ
angle.
.
sin IJ cot IJ + cas
16. Fmd the value of
see IJ co t IJ
acute angle.
.
sin IJ
sin' IJsee
17. Fmd the value of
cas IJ + cas' IJ tan'
acute angle.
IJ
'
V1O.
= 10'
and IJIS an acut,e
Am. ~o..
.
when cot IJ = v- r;;5, and IJIS
an
IJ
IJ'
when cse IJ
=
Ans. 0.745.
.
3 and IJ IS an
Ans. 1.414.
27. Trigonometric functions applied to right triangles.-When
the angle 8 is acute, the abscissa, ordinate, and distance for
any point in the terminal side form a right triangle, in which the
given angle 8 is one of the acute angles. On account of the many
applications of the right triangle in trigonometry, the definitions
of the trigonometric functions will be stated
B
with special reference to the right triangle.
These definitions are very important and are
c
a
frequently
the first ones taught, but it
A.f}
should be carefully noted that they are not
c
0
b
general because they apply only to acute
+d
FIG. 27.
terminal side, as in Fig. 28.
and CA = b.
By definition:
sin A = ordinate
distance
side opposite.
= !!:
c = hypotenuse
cos A = abscissa = ~ = side adjacent.
distance
c
hypotenuse
ordinate
!!: side opposite.
tan A =
abscissa = b = side adjacent
cot A = abscissa = !! = side adjacent.
ordinate
a
side opposite
distance
hypotenuse.
~
sec A =
abscissa = b = side adjacent
csc A = distance = ~ = hypotenuse.
ordinate
a
side opposite
Again, suppose the triangle ABC placed so that LB has its
vertex at the origin, BC for the initial side, and BA for the
OF ONE ANGLE
The coordinates
of A are BC
sin B = ~ =
c
hypotenuse
side opposite.
side adjacent.
cot B = !!:
b = side opposite
cos B = !!: = side adjacent.
c
hypotenuse
tan B = ~ = s~de opposite.
a
sIde adjacent
~ypot~nuse.
sec B = a!: = sIde
adjacent
29
= a
~ypotenu~e.
csc B = b~ = sIde
OpposIte
Then, no matter where the right triangle is found, the functions
of the acute angles may be written in terms of the legs and the
hypotenuse of the right triangle.
y
/fA
/
(b)
B
b
angles.
Draw the right triangle ABC (Fig. 27),
with the vertex A at the origin, and AC on the initial line. Then
AC and CB are the coordinates of B in the terminal side AB.
Let AC = b, CB = a, and ,iB
=~_n
:lty defiJiition:
FUNCTIONS
A~~J
"~:
c-x
C
B
A
b
1 \ (rI)/
'B
FIG. 29.
FIG. 28.
B
I
')C
'A
EXERCISES
1. Gin, ~)ra!i:v-r.hC'six trigOJl(JIIletrigmtios
of C'Rclu:lf th",
'''''It<. angles of
the right triangles in Fig. 29.
In the right triangle ABC, find the six trigonometric
ratios from the
following data:
2. a = ie.
3. b = lc.
5. In a right triangle find a if sin A
6. In a right
triangle
find b if eos A.
triangle
find a if cot A
triangle
find
e = 53.16.
7. In a right
b = 18.7.
_8.
In
a right
4. a = 4b.
= I, and e = 4.28.
= i,
Ans. a = 2.568.
and
Ans. b = 17.72.
Ans.
e if sin
a - 12.65.
Ans.
9. In a right triangle find a if tan
b = 8.32.
Ans.
10. In a right triangle find b if cot B
11. In a right triangle
find a and e if
12. In a right triangle
find a and b if
~
B
= t, and
a = 11.22.
C
a
A
/0, and
e - 40.48.
A
C
B
b.
= 7.5, and
FlU. 30.
a = 1.109.
4.56,
and
a
=
= 42.
Am. b = 9.21.
sin B
= i, and b = 22.45.
Ans. e = 33.675; a = 25.099.
sin A
= 0.236, and e ~ 45.
Ana. a = 10.62; b -- 43.73.
::
-.
30
PLANE
The following
13. c
AND SPHERICAL
TRIGONOMETRY
TRIGONOMETRIC
Ans. tan A =
~
2rs
rZ
+ s'
Ans. sin A = v'2Ts
~,
c = r +
r + s'
2rs
15. a = 2rs, b = rZ - sz; find cos B.
Ans. cos B = rZ
+ s'
16. Construct a right triangle in which sin A
= 2 sin B. In which sin
14. b =
s; find sin A.
A = 3 cos A. In which tan A = 3 tan B.
Construct the angle IJ from each of the following data:
17. tan (J = 2 cot (J. 19. sin IJ = 3 cos IJ.
21. cot (J = 3 tan (J.
18. cos (J = 2 sin IJ. 20. sec (J = 2 csc (J.
22. sin IJ = cos (J.
28. Relations
between
the functions
of complementary
angles.-From
the formulas of Art. 27, the following relations
are evident:
b
sin A = cosB =~.
cot A = tan B = -.
c
a
cas A = sin B = ~.
c
~.
tan A = cot B =
b
sec A = csc B = ~.
b
csc A = sec B = £..
a
But angles A and B are complementary;
therefore, the sine, cosine,
tangent, cotangent, secant, and cosecant of an angle are, respectively,
the cosine, sine, cotangent, tangent, cosecant, and secant of the complement of the angle.
They are also called cofunctions.
For example, cas 75° = sin (90° - 75°) = sin 15°;
tan 80° = cot (90° - 80°) = cot 10°.
1. Express
the following
EXERCISES
functions as functions
of the complements
()
=
Take
11.
r
r = 5 units.
y
If
If
If
If
If
90°).
sin 40° = cos 0, find 0.
tan 50° = cot 21J, find IJ.
csc 20° = sec 21J, find IJ.
cos (J = sin 21J,find IJ.
cot ilJ = tan IJ, find IJ.
6.
7.
8.
9.
If
If
If
If
sin
tan
csc
cos
21J = cos 41J,find IJ.
IJ = cot 51J,find IJ.
61J = sec 41J,find IJ.
llJ = sin (J, find IJ.
A:"s.67!0.
3
= 3 units and
Draw
AB "OX
3
4
M2
4
MI
and 3 units above it as in Fig. 31.
,X
FIG. 31.
Construct the arc of a circle with
center at 0 and radius 5 units, intersecting AB at PI and P2.
Then for PI, x = 4, y = 3, and r = 5; for P2, x = -4, y
= 3,
and r = 5. Now OPI and OP2 are terminal sides, respectively of
LXOPI = ()l and LXOP2 = ()2, each of which has its sine equal
to i. Then from the definitions of the trigonometric functions
we have the following:
I
Quaurant
Angle
I
)
'H
II.
sin IJ
I
.
IJ,
"""'"
..........
IJz
Example
B
~V
2.-Given
IY
I
I
cos 0
I
i
3
Ii
3
4
Ii
-t
Ii
I
cot 0
tan IJ
I
I
.-'
I
.I
i
i
3
"'3
-"4
sec IJ
4
S
-!
csc (J
I
I
5
4
-i
jf15
i
cos (J = -j.
Construct (Jand find all the
other functions.
Solution.-By
definition,
cas (J =
x
2
.
.
= -3' Smce r IS a 1ways pas 1.t Ive,
'
r
0
of
these angles: sin 60°; cos 25°; tan 15°; cot 65°; sec 10°; csc 42°; sin 0; sin 31J;
cos (0
2.
3.
4.
5.
10.
31
29. Given the function of an angle in any quadrant, to construct the angle. Example I.-Given
sin () =~.
Construct
angle () and find all the other
y
functions.
P2/
IH
Solution.-By
definition, sin A
I..
Note.-Theterm
cosine was not used until the beginning of the seventeenth
century. Before that time the expression, sine of the complement (Latin,
complementi sinus) was used instead. Cosine is a contraction of the Latin
expression. Similarly, cotangent and cosecant are contractions of complemenli tangens and complementi secans respectively.
The abbreviations, sin, cos, tan, cot, sec, and csc did not come into general
use until the middle of the eighteenth century.
OF ONE ANGLE
11. If cos IJ = sin (45° - !1J)rfind (J.
Am. 90°.
12. If cot a = tan (45° + a), find a.
Am. 22° 30'.
13. If csc (60° - a) = sec (15° + 3a), find a.
Ans. 7° 30'.
14. If sin (35° + (J) = cos ({J- 15°), find {J.
Ans. 35°.
15. Express each of the following functions as functions of angles less
than 45°: sin 68°; cot 88°; sec 75°; csc 47° 58' 12"; cos 71° 12' 56".
refer to a right triangle:
= r + s, a = v'2Ts; find tan A.
FUNCTIONS
FIG. 32.
angles.
The functions
we take x = -2 units and r = 3
oX units. Draw AB
OY and 2 units
to the left as in Fig." 32. Construct
a circle of radius 3, with its center at
0, and intersecting AB at PI andP2.
Draw OPI and OP2. As in Example
1, it may be shown that LXOPI
=
(Jl and LXOP2 = (J2are the required
are as follows:
l
32
PLANE
Angle
Quadrant
TRIGONOMETRY
AND SPHERICAL
cos 9
sin 9
~an9
I
0
.
0,
III. . .. . .. . . . .
0,
II...........
Example
cot 9
2
-3"
tan ()
the other functions.
Solution.-By
tan
2
0
2
II...
0
3
3
-0
angle () and find all
Hence
=x
3
-2
0
Construct
()
3
-2
-0
2
= i.
definition,
I
0
2
I-f
3.-Given
-2
-3"
3"""
0
see 0 lese
I
I
TRIGONOMETRIC
_3
'!f..
-3
x = 4 = - 4'
and we may take y = ::!::3and x = ::!::4. Then
r = y(::!::4)2 + (::!::3)2= 5.
II
1. cos 0 = -to
2
9. csc 9 = 0'
10. sin 9 = -~.
5. cos 9 = 0.6.
2. sin 9 = -1\'
6. cot 0 = 3.
3. tan 9 = -i.
7. tan 9 = -0.
11. tan 0 = ~.
4. sin 0 = 'J'D'
8. see 9 = -4.
12. csc 0 = 2.4.
13. What is the greatest value that the sine of an angle may have?
The
least value?
How does the value of the sine change as the angle changes
from 0° to 90°? From 90° to 180°? From 180° to 270°? From 270°
to 360°?
14. Answer the questions of Exercise 13 for the cosine.
For the tangent.
In Exercises 15, 18, and 21 show by substitution
that the right-hand
member is equal to the left.
(1
+
tan2
0) (1 -
cot2
9)
=
in the second quadrant.
1.6 F'IIId t h e va Iue 0f
sec2 9 -
sin 0 tan 9
see 0
csc2
9, when
sin 0 =
1.7
fourth
18.
F'm d t h e va Iue 0 f
+
vel'S
0 cot
0
=
w h en csc 0
0'
5
= -4
quadrant.
Cos
9 tan
in the fourth
0
+
cos 0, when
quadrant.
~eeollJ ljuadrant.
.
Fmd
sin
of
see 0 - cse 0,
when tan 0
see 0 + CRC IJ
-2
respectively of OY.
quadrant.
Also draw EF and GH
II
OX and 3 units
above and below OX respectively.
These lines and the circle
intersect at the points PI, P2, Pg, and P4. Since x and y must
both be positive or both negative, the required points must be PI
and Pg located in the first and third quadrants.
Draw OP1 and
OPg forming the angles XOP1 = (}1 and XOPg = (}g. The
functions are as follows:
Quadrant
Angle
I
1............
III. . .. . . . . . . .
sin 0
cos 9
I
0,
0,
tan 9
I
t
!
-f
-i
3
"4
3
see 9
cot 0
I
I
"4
I
csc 0
I
I
t
I
t
5
.{
-1
I
i
-J
0 is
and (J is in the
Ans. 3.
the value
21. cot 0 +
0 is
-2.133.
see IJ = 2 and
of
sin 0 + cot 0
, when
cosO+cscO
. /0
cot 0 = 2v
2 and
Ans.
FIG. 33.
and
Ans. 1"..
..
an d 0 1S m t h e
Ans.
0 + sin
19. Find the value
20.
sin 9 + tan 9
cos 0
t
..
cot 0 = - 23 an d 0 1SIII
t h e f ourt h
,wen
h
quadrant.
x
33
OF ONE ANGLE
EXERCISES
Draw the angles less than 360° and tabulate the six trigonometric ratios
determined by each of the following:
15.
With 0 as a center and 5 as a radius, construct a circle as in
Fig. 33. Draw AB and CD OY and 4 units to the right and left
FUNCTIONS
1 ~~:s IJ =
csc 0, when sin 0 = -
f
.
1
sm 0 = -3'
-0.6328.
and 0 is in the third
l
RELATIONS BETWEEN
[6]
cot 6
TRIGONOMETRIC
= -tan1 6
and
tan 6 =
FUNCTIONS
35
1
-.
cot 6
The following formulas are easily derived:
CHAPTER
RELATIONS
BETWEEN
III
TRIGONOMETRIC
FUNCTIONS
30. Fundamental
relations between the functions of an
angle.-In
handling questions that occur in mathematics
a
great deal of use is made of relations that exist between trigonometric functions of angles. These relations are numerous,
but it is necessary to memorize only a few of them. In this
chapter are considered only those relations that exist between
functions of one angle. In a later chapter will be found relations
where different angles are involved.
From the figures of Art. 15, it is evident that for an angle in any
quadrant
(1)
x2
x2
Dividing
But::
(1) by r2, 2
r
r = cos
[1]
+ y2
y2
+ 2r
But
tan
r2
() and 1f.
r = sin ().
. . . sins 6 + coss 9
()
sec ()
1f. and
= x
. '. 1
2
(1) by y2, X2
But cot ()
y
and csc
= ::
y
= 1.
mi-------
[5]
=
-J-
The eight formulas of this article are identities, for they are
true for any angle whatever.
They are often spoken of as fundamental identities,
or formulas.
They should be carefully
memorized as they are frequently used.
It will be noted that throughout the book the important
formulas are printed in bold-faced type and numbered in square
brackets for ready reference.
The following examples make use of the fundamental formulas
in computing the other trigonometric functions when one function is given. Compare the work with that of the previous
articles where the angles were first constructed.
() in the first quadrant,
Example I.-Given
tan ()
By [6],
By [3],
By [4],
2
By [5],
= !.
Y
and
~.
csc 9
sin 6
=
cas 9
= -. 1
sec 6
by means of the fundamental
-
+ 1 = yr2'
sm 6
1
sec 6 = and
cas 6
34
the other functions
------
= !.
x
. . . 1 + cotS 6 = cscs 6.
Also, from the definitions of the trigonometric
following reciprocal relations are evident:
csc 6
6
6'
6
6'
-L
--
[3]
[4]
[8]
determine
formulas.
+ tanS 9 = secs a.
()
tan6 =
= t, and
(1) by x2, l--1-Y: ~--r2'
x
x
[2]
Dividing
r2.
= "2r = 1.
~
Dividing
=
sin
cas
cas
cot 6 =
sin
[7]
functions,
the
cot
csc
()
()
= tan
-
1
= VI
()
=
+
1
-
t
cot2
=
3
-.
4
()
= VI
+
1\
= t.
~
~ =~.
sin (J =
csc () = t
5
113
()
cas = - () = - = -.
sec
t 5
Example 2.-Given
sin () = !, and () in the second quadrant,
determine the other functions by means of the fundamental
formulas.
Solution.-By[1],cos()=
-Vl-sin2()=
-Yl-t= -h/3.
()
!
1
By [7],
t an () = sin () =
= -3v - r3.
cas
-!V3
By [6],
11cot (J =
tan () = -tv3
= -v3.
-
36
PLANE AND SPHERICAL TRIGONOMETRY
By [6],
1
sec 0 =
1
RELATIONS BETWEEN
2-
By [4],
=
. cos 0 _.1-2VU "3 = -3V3.
1
1
csc 0 = ---;-- = - = 2.
Note.-The
proper algebraic sign is determined
.'. sin 0 = VI - cos2 0 =
By [2],
!
sm 0
by Art. 17.
In Exercises 1 to 10 determine the remaining functions from the given
functions by means of the fundamental identities and check by constructing
the angle and computing the functions.
1. Given sin IJ = -i, and IJin the third quadrant.
2. Given tan IJ = -,"", and IJin the fourth quadrant.
3. Given sec IJ = V2, and IJin the first quadrant.
4. Given cos IJ = V3, and IJin the fourth quadrant.
.
Given
Given
Given
Given
Given
Given
tan
cot
sec
tan
csc
sin
2
IJ =
IJ =
a =
fJ =
IJ =
a =
!, and IJin the third quadrant.
5, and IJin the first quadrant.
-!, and a in the third quadrant.
~l, and ,8in the third quadrant.
-V, and IJin the fourth quadrant.
-..91' and a in the third quadrant.
11. If cos !a = -f8(8=a)
~,
°'1
a+b+e
=
where
8
where
8 =
?
,show
.
that
sm
!a
that
tan
h
1(8- b)(8 - e).
= -'1
be
12. If cos h
e)
/8(8
= "\.
,,~' -
a + b + e
,show
2
1(8 a)(8
= '1 -8(8 - e)- b).
13. If tan fJ = e, show that csc fJis real for all values of c.
-
14. G Iven
sm 'Y
"
2mn
= ~m + n ,;
s h ow t h at tan
2mn
'Y
= :!:~.
m -n
31. To express one function in terms of each of the other
functions.-Any
trigonometric function can readily be expressed
in terms of any other function by means of the fundamental
formulas.
While the work cannot be carried out so rapidly as
by the method of the following article, it gives needed drill in
the use of the formulas.
Example.-Express
sin 0 in terms of each of the other functions.
By [1],
sin '0
By [6],
cos 0 =
= VI - cos2 O.
-
1
sec 0
'
sec2 0 = I
+
11
~
-
~
sec2 0
=
37
vsec20
- 1.
sec 0
tan2 O.
vsec20 - 1
vtan20
tan 0
sec 0
VI + tan2 0
vi + tan2 0
1
tan 0
cot 0
1
Also sin 0 =
=
VI + tan2 0 = 11+~
VI + cot2 0
'\J
coV 0
1
By [4],
sin 0 =
csc O'
.'. sin 0 =
EXERCISES
5.
6.
7.
8.
9.
10.
TRIGONOMETRIC FUNCTIONS
The algebraic sign of sin 0 is determined from the quadrant in
which 0 is found.
32. To express all the functions of an angle in terms of one
function of the angle, by means of a
y
triangle.-The
scheme outlined in this
article can be carried out rapidly and
P
will be found of very great use in future
work.
Example I.-Express
all the functions
01
M X
of 0 in terms of sin O.
'./I-sin21J
Solution.-Construct
angle 0 in the
first quadrant
(Fig. 34) and choose
the point P in the terminal side with coordinates
OM
and MP.
Then,
by definition,
sin 0 =
~:'
and, if OP ii>
taken equal to 1, MP = sin 0, and OM = VOp2 - MP2 =
VI - sin2 O.
The remaining functions may then be wr}tten as follows:
OM
.
OP
1
cos 0 = = VI - sm2 O. sec 0 = =
.
OP
OM
VI - sin2 0
MP
sin 0
OP
1
tan (J = ~.
OM = vI - sin2 (J csc 0 = -MP = sm (J
OM
VI - sin2 0.
co t 0 MP sin (J
Example 2.-Express
all the functions in terms of cos O.
Solution.-Construct
angle 0 in the first quadrant (Fig. 35)
and choose the point P in the terminal side with coordinates
38
RELATIONS BETWEEN
PLANE AND SPHERICAL TRIGONOMETRY
OM and MP.
Then, by definition,
y
cos (J =
~~, and, if OP is
equal to 1, OM = cos (J, and
MP = V OP2 - UM2=VI
- cos2 (J.
~;
sin (J =
FIG. 35.
MP
eos 0 IVI
-
sin'
VI
VI
/
'O=-=O
- sin'
. .. I.
"n
-
~in-IJ
see 0 I
cae (J
VI
lv/l
1
VI -
I VI
VI
-
1
VI
-
vsec'
1
+
see 0
cot 0
vi-+
cot' 0
tanie
~cot
0
0
vsee'
0 -
1
llill fJ
-
VI
+ tan'
0 VI
VI
+ tan'
0
+
cot ,- 0
!V""""
0
0
tan
0
~
-
1
ese 0
~
1
cse 0
---==
1
vese'
definition
sin (J
-cos (J' 1 + tan2
and
~a::~
(1
(J)2
in
(J
covers
=
1 -
sin
(J,
1
-sec
(J'
these values, we have
sin4 (J
cos4 (J
1
2' sm 2 (J
sec :! (J = -
2 sin2 (J + sin4 (J = (1
-
formulas,
(J = sec2 (J, and cos (J =
1 - 2[1 - (1 - sm (J))2+
sin4 (J
+
cos4 (J
-y-
cos4 (J
-
sin2 (J)2
(cos2
(J)2
=
cos4 (J.
EXERCISES
Transform the following expressions as indicated:
1. sin 0 cot 0 see 0 to 1.
2. cos 0 tan 0 csc 0 to 1.
3. (1 - sin2 </»(1 + tan2
to 1.
0
-
1
see 0
6.
7.
cse 8
The table has been prepared under the assumption that (Jis an
acute angle. Should (Jbe in any other quadrant, the proper sign
for each function may then be determined.
33. Transformation of trigonometric expressions.- In all transformations, avoid radicals if possible. Usually, this can best be
done by changing to sines and cosines and then simplifying.
It
will be noticed that, if there are no radicals in an expression, it
can be changed to sines and cosines without using radicals.
If
the expression is in a factored form, it is often desirable to reduce
each factor separately and multiply the results.
cos (J .
Example I.-Express
.
m terms of tan (J.
SIn (Jco t2 (J
0) vers 0 + covers 0 (1 + sin 0) to 1.
1>
- :~(>('2
-,
to tan' rb.
.
1 - csc</>
see 0 - tan 0 sin 0 to cos o.
sin2 </>(1+ sec2 </» to sec2 </> cos2 </>.
sin 0
to cot 0 + csc o.
1 cos 0
sin4 </>- cos4 </>to 1 - 2 cos2 </>.
sin40 + cos4 0 to 1 - 2 sin2 0 cos2 O.
.
sm 2 </>see </>
4. (1
-
'"'
see 0
covers (J)2 +
</»
vese'
0- 1
,
:---
VI + cot' 0 VBee' 0 - 1
I
~
1
1 - 2(1 -
.
-"se.~
V ese'
0 -
1
see 0
1
1
0- 1
cot 0
I
eos'
1
+ tan'-II VI + cot' 0
tan
0<-"'"8-
eos 0
0
1
SiilO
0
- eos' 0
eos 0
cos 8
~
sin'
eos'
cos 8
0
sin
tan 0 I
-
tan 0
2.-Express
Substituting
= 1
~in (J
cos (J
cos2 (J = COS(J = tan (J.
.
sm (J 0
sin2 (J
of cos (Jo
tan (J =
In the following table, the student is asked to show that each
function in the first column is equal to every expression found in
the same row with the function:
sin (J
Example
Solutiono-By
VI
OM -
=
terms
- cos2 (J.
= VI
OP
1
- cos2 (J
sec (J =
cos (J
OM --. cos (J
OM
cos (J
OP
1
cot (J =
csc (J =
MP - VI - cos2 (J
MP - VI - cos2 (J
sin 8
cos (J
cos (J
(J = ---o--(J then .
SIn '
sm. (J co t 2 (J
Solutwn.-cot
taken
The remaining functions may then be
written as follows:
tan (J =
.
39
TRIGONOMETRIC FUNCTIONS
8.
9.
10.
11
.
+ cos
-
-
1
+ see
</>
+ cos
</>t 0
1.
o I cos~0
12. cos2\/o~ 11 +cossin'
0
"\/ 1 + sm2
[
~
~
I
see'
cf>-
1
coP
-
14. 7 see' </>
6 tan2 cf>+ 9 cos2 cf>to
+
cf>- 1
to (1
ese' cf>
cf>. lese'
"\j sec2 cf>(1
+ csc cf>\/
+ cot' cf»
cos cf».
13.
11+ sin2 O
cos 0 J
+ -\/
~
0
(1
+
to 1.
+ cot
cf»(1
-
sin cf>
3 cos' cf»' .
COS 2 cf>
+ 2 cos' cf>+ sin' cf>to 3 + tan2 </>.
1 - tan' cf>
1 - ta114 cf>
(1 - vers' 0)2 - (1. - covers' 0)'
16.
to 5(cos 0 + sin 0) -4(1 +sin Ocos 0).
cos 0 - sm 0
16.
sin2 cf>cos2 cf>
cos4 cf>
/
40
PLANE AND SPHERICAL TRIGONOMETRY
34. Identities.When two expressions in some letter x are
equal for all values of that letter they are said to be identically
equal.
The equation formed by equating the two expressions is called
an identity.
The symbol denoting identity is ==. When there can be no
misunderstanding as to the meaning, the sign of equality is often
used to denote identity.
The symbol == is read" identically
equals," or "is identically equal to."
Thus, x2 - 1 == (x - 1) (x + 1) because the equation is true
for all values of x.
Since the fundamental formulas are true for all values of (J,
they are identities.
In showing that one trigonometric expression is identically
equal to another, we either transform both expressions to the
same form, or transform one expression into the other, by means
of the fundamental formulas.
That is, if A is to be proved
identically equal to B, it can be done by
(1) Changing A to B,
(2) Changing B to A, or
(3) Changing both A and B to a third form C.
In the applications of this part of trigonometry, however,
one usually knows exactly into what form a certain expression
mu~t. be tram,formerl.
For thi~ rea~on if i~ u,"ual to refluire the
student to change the first member of an identity into the second.
It is usually best, especially for the beginner, to express all the
functions of the expression which is to be transformed in terms
.
of sine and cosine before attempting to simplify.
Avoid radicals whenever possible.
When the expression that is to be transformed is given in a
factored form, it is usually best to simplify each factor separately
before multiplying them together.
Example I.-By transforming the first member into the second
prove the identity tan (Jsin (J + cos (J = sec (J.
sin (J
Proof.-Substituting
for tan (J,we have
cos (J
sin2 (J + cos2 (J
1
sin (J . . (J
SIn
= = sec. (J
+ cos (J =
cos (J
Example
prove
cos (J
cos (J
2.-By transforming
th e I'd en t'tI y
the first member into the second
cot ex cos a
cot a - COSa
cot a + cos ex
=
cot a cos a
RELATIONS BETWEEN
C?Sa for cot a, we have
Proof.-Substituting
SIn a
COS a
sma
cos a
+
cos2 ex
;--
. COS a
;
cot ex COSa
cot ex
cos a
41
TRIGONOMETRIC FUNCTIONS
sma
- cos a(1
+ COSa
;
SIn a
cos2 a
- cos a(1
+
.
+ sin
a)
SIn a
1 - sin2 a
sin a) = cos a(1
+
sin a)
=
1 - sin a
COS
ex
Now multiply the numerator and denominator by cot a, and we
have
1 - sin ex . cot a
cot a - sin a cot ex
cot a - COSex
-cot a
COSa
cos a cot ex
COSex cot a
EXERCISES
Prove the following identities by transforming
identity into the second:
1
. cas cot8 csc8
8
= 1
the first member of the
.
2. tan 8 cas 8 = sin 8.
3. sec 8 cot 8 = csc 8.
sin 8 sec 8
4.
= 1.
tan 8
6. (1 - cos2 <1»sec2 <I>= tan2
6. sec2 <I> csc2 <I>= sec2 <I>csc2 <1>.
1,
1
~
fo;}Tl"
<1>.
+
(
)
'
7. :;-~ln"'...
1> =
.
'P'
cot'
cot- <I>
' <I>
8. cot2 <I>- cos2 <I>= cos2 <I>cot2 <1>.
9. (sec2 8 - 1)csc2 8 = sec2 8.
'
10. cot 8 + tan 8 = cot 8 sec2 8.
11. (tan <I>+ cot <1»2 = sec2 <I>csc2
12. (cas 8 - sin 8)2 + 2 sin 8 cas 8
= 1.
<1>.
13. sec44>
16.
16.
17.
18.
4> = (sec2 4»(2 sin2 4> + cos2 4».
tan4
-
14. tan 8(sin 8
1 + csc 8
+
cas 8)2 cot 8
1+
-
2 sin 8 cas 8
= 1.
8.
csc 8 - 1 = 1 - sm 8
sin fJ Vsec2 4>- 1 . (1
- sin2 4»-' = tan 4>tan fJ.
see 4>v. / 1 - sin2 fJ
cos.8 _1 - sin 8
2 tan 8.
1 - sm 0
cas 8 =
(1 - tan 4»2
sec 2 4> + 2 sin 4>cas 4>= 1.
19. s~n 8
+
s~n 4>
s~n
csc 4> + csc O.
sm 8 - sm 4>= esc 4>- csc 8
(1 + sin 4»
/1 - s~n4>
/1
20.
+
["J1+sm4>
~
2cos</>
~
+ s~n</>
-"I-sm</>
J
~1=S~n
</> sec </>.
=
l+sm<l>
/
40
PLANE AND SPHERICAL TRIGONOMETRY
34. Identities.When two expressions in some letter x are
equal for all values of that letter they are said to be identically
equal.
The equation formed by equating the two expressions is called
an identity.
The symbol denoting identity is ==. When there can be no
misunderstanding as to the meaning, the sign of equality is often
used to denote identity.
The symbol == is read" identically
equals," or "is identically equal to."
Thus, x2 - 1 == (x - 1) (x + 1) because the equation is true
for all values of x.
Since the fundamental formulas are true for all values of (J,
they are identities.
In showing that one trigonometric expression is identically
equal to another, we either transform both expressions to the
same form, or transform one expression into the other, by means
of the fundamental formulas.
That is, if A is to be proved
identically equal to B, it can be done by
(1) Changing A to B,
(2) Changing B to A, or
(3) Changing both A and B to a third form C.
In the applications of this part of trigonometry, however,
one usually knows exactly into what form a certain expression
mu~t. be tram,formerl.
For thi~ rea~on if i~ u,"ual to refluire the
student to change the first member of an identity into the second.
It is usually best, especially for the beginner, to express all the
functions of the expression which is to be transformed in terms
.
of sine and cosine before attempting to simplify.
Avoid radicals whenever possible.
When the expression that is to be transformed is given in a
factored form, it is usually best to simplify each factor separately
before multiplying them together.
Example I.-By transforming the first member into the second
prove the identity tan (Jsin (J + cos (J = sec (J.
sin (J
Proof.-Substituting
for tan (J,we have
cos (J
sin2 (J + cos2 (J
1
sin (J . . (J
SIn
= = sec. (J
+ cos (J =
cos (J
Example
prove
cos (J
cos (J
2.-By transforming
th e I'd en t'tI y
the first member into the second
cot ex cos a
cot a - COSa
cot a + cos ex
=
cot a cos a
RELATIONS BETWEEN
C?Sa for cot a, we have
Proof.-Substituting
SIn a
COS a
sma
cos a
+
cos2 ex
;--
. COS a
;
cot ex COSa
cot ex
cos a
41
TRIGONOMETRIC FUNCTIONS
sma
- cos a(1
+ COSa
;
SIn a
cos2 a
- cos a(1
+
.
+ sin
a)
SIn a
1 - sin2 a
sin a) = cos a(1
+
sin a)
=
1 - sin a
COS
ex
Now multiply the numerator and denominator by cot a, and we
have
1 - sin ex . cot a
cot a - sin a cot ex
cot a - COSex
-cot a
COSa
cos a cot ex
COSex cot a
EXERCISES
Prove the following identities by transforming
identity into the second:
1
. cas cot8 csc8
8
= 1
the first member of the
.
2. tan 8 cas 8 = sin 8.
3. sec 8 cot 8 = csc 8.
sin 8 sec 8
4.
= 1.
tan 8
6. (1 - cos2 <1»sec2 <I>= tan2
6. sec2 <I> csc2 <I>= sec2 <I>csc2 <1>.
1,
1
~
fo;}Tl"
<1>.
+
(
)
'
7. :;-~ln"'...
1> =
.
'P'
cot'
cot- <I>
' <I>
8. cot2 <I>- cos2 <I>= cos2 <I>cot2 <1>.
9. (sec2 8 - 1)csc2 8 = sec2 8.
'
10. cot 8 + tan 8 = cot 8 sec2 8.
11. (tan <I>+ cot <1»2 = sec2 <I>csc2
12. (cas 8 - sin 8)2 + 2 sin 8 cas 8
= 1.
<1>.
13. sec44>
16.
16.
17.
18.
4> = (sec2 4»(2 sin2 4> + cos2 4».
tan4
-
14. tan 8(sin 8
1 + csc 8
+
cas 8)2 cot 8
1+
-
2 sin 8 cas 8
= 1.
8.
csc 8 - 1 = 1 - sm 8
sin fJ Vsec2 4>- 1 . (1
- sin2 4»-' = tan 4>tan fJ.
see 4>v. / 1 - sin2 fJ
cos.8 _1 - sin 8
2 tan 8.
1 - sm 0
cas 8 =
(1 - tan 4»2
sec 2 4> + 2 sin 4>cas 4>= 1.
19. s~n 8
+
s~n 4>
s~n
csc 4> + csc O.
sm 8 - sm 4>= esc 4>- csc 8
(1 + sin 4»
/1 - s~n4>
/1
20.
+
["J1+sm4>
~
2cos</>
~
+ s~n</>
-"I-sm</>
J
~1=S~n
</> sec </>.
=
l+sm<l>
r
J/
42
PLANE AND SPHERICAL TRIGONOMETRY
RELATIONS
21. (tan' 6 + l)eot' 6 = ese' 6.
22. sin' <I>see <I>(sin' 6 see 6 + eos 6) + eos <I>(sin' 0 see IJ + eos 6) =
see 6.
23. 2 sin <I>eos <I> sin' <I>tan <I> eos' <I> eot <I> = see <I> ese <1>.
<I> - 1)
24. sin <I>eos <I>[2 + (see'
+ (ese' - 1)] = see <I>ese <1>.
25. eos 6 (see 6 + ese 6) + sin 6 (see 6 - ese 6) = see 6 ese 6.
1 + eos
see
<I>
+
+
<I>
~-
26.
1
<I>
eos
<I>
= ese
+ eos
see'
<1>(1
1 -
3 cos'
<I>
<I>
tan
+
eot
<1>.
29.
<I>sin'
+2
sin'
<I>
<I> =
<I>cos'
(sin'
vers B covers
B(cos
B -
+ cos'
sin B)(1
+
<1»'.
sin B
+
eos B).
31. (2 sin' 0 - cos' 6)' - 9(2 sin' 6 - I)' =
32.
33
.
tan'
(2 - 3 sin' 6)(2
<I>(see <I>- 1)
- see' <I>= 1 - 2 see <1>.
see <I>+ 1
VI
- sin <I>eos <1>
sin
<I>cos
<I>
[
1
J+
.,
sm
+
cot
<I>
+ eos'
tan
<I>
I
+
-'
<I> "'\J ese'
<I>
1
3 sin6) (3 sin 6 -
+ cot'
<I>
<1>(1+ cot <1»
]
2).
= 1.
6]
cos6 6
34. sin' 6lcos. 6 - sin'
= see' 6.
eos' 6[2 cos' 6 - 1]
tan a + tan {3 see a + see {3.
35.
tan {3
see a
see {3 = tan a
-
-
36. Inverse trigonometric
functions.-The
FUNCTIONS
43
be taken to mean
~
= csc a, which i~something entirely differ-
sma
ent from our meaning of sin-1 a as explained at the beginning of
this article.
that
sin cos-1
H = Ts",
Solution.-Let
(J = cos-1 H.
Then from the definitions of the inverse functions,
<I>
30. eovers B(1 - cos' B) - vers B(1 - sin' B) =
TRIGONOMETRIC
(sin (J)2and x-I for!' and so the symbol sin-1 a might consistently
x
Example.-Show
!.
<1»
(tan <I>+ see <1»'+ 1 = 2
28. 2 tan' 6 + 2 tan 6 see 6 + 1 = see' 0(1 + sin 6)'.
27.
BETWEEN
equation
sin (J = a
means that (Jis an angle whose sine is a. The expression sin-1 a
is an abbreviation for the expression" an angle whose sine is .a."
Then we may write
(J = sin-1 a.
The form sin-1 a is also read "anti-sine a," "inverse-sine a,"
"arc sine a." It is also written invsin a and arc sin a.
Analogous forms with analogous meanings are given for the
other functions.
Illustrations.-sin-1
t = 30 or 150°. cos-11 = 0°. tan-II =
45 or 225°.
The notations sin-1 a, cos-1 a, etc. have the advantage that
they are the forms most frequently used in other branches of
mathematics and its applications; but they have the disadvantage
of conflicting with the customary notation for exponents, and so
tend to cause confusion. Thus, sin2(J is usually written for
cos (J = H,
cos2
(J = VI - (HV = Is".
.
. . sin cos-1 H = Ts".
This could also be solved by constructing the angle.
By [1],
sin (J = VI
EXERCISES
Answer Exercises 1 to 12 orally, considering only angles that are less
than 900.
.
1 sm cos -1
.
V2
5. eos sec.-1 5
2'
9. sin sec-1 H.
2. sin sin-1 ~1~.
3. tan sec-1 2.
4. cos ese-1 3.
Prove the relations
6. tan sin-1 H.
10.
7. sin eos-1 O.
11.
8. sin tan-1 0.
12.
in Exercises 13 to 22.
18. cas sin-1 a
U, = ::f:V'I - u,2.
cse eot-1 l.
sin cos-1 t.
cos sec-1 5.
13. Sill eus'l
= :+:~/ I - n'.
14. sin tan-1 a = ::f:va.
1
+ a'
19. eos tan-1 a = ::f:V~.
1
+ a'
15. sin cot-1 a = :1:V~.
1 + a'
16. sin sec-1 a = + V a' - 1.
a
20. cos cot-1 a = :1:V~.
1
1
21. eos sec-1 a = -.
a
+ a'
.
1
vaz=I
17. sm csc-1 a = -.
22. COScsc-1 a
= :1:
.
a
a
For angles not greater than 900, show that the following are true:
23. sin-1 H = COS-1i~.
24. tan-1 H = sin-' {~.
36. Trigonometric
equations.-A
trigonometric
equation
is
an equation in which the unknown is involved in a trigonometric
function.
The solution of a trigonometric equation is a value of the angle
which satisfies the equation.
In general, both algebra and trigonometry
are involved
in
solving a trigonometric
equation.
Algebra must be used when
the trigonometric
functions are involved algebraically
in a trigo-
44
RELATIONS BETWEEN
PLANE AND SPHERICAL TRIGONOMETRY
no metric equation, for then the
some trigonometric function.
Thus, sin2 8 - t sin 8 + ! =
and, algebraically, is solved for
is solved for x, either by the
equation or by factoring.
The
equation must first be solved for
0 is a quadratic equation in sin 8;
sin 8 exactly as x2 - ~x + ~ = 0
formula for solving a quadratic
solutions for sin 8 are
Example
I.-Solve
sin 8 =
!V2 for 8 < 90°.
Here all that is necessary is to know the angle less than 90°
whose sine is !V2. From the table on page 24 this is found to
be 45°,
.'. if sin (J = !V2, (J = 45°.
Example 2.-Solve tan (J = 0.43654 for (J < 90°.
This value of 8 cannot be found by referring to page 24, as it
requires a more extensive table of natural functions.
By referring to Table IV, (Jis found to be 23° 35'.
. . . if tan 8 = 0.43654, (J = 23° 35'.
Example 3.-Solve cos (J = 0.77467 for (J < 90°.
From Table IV, (Jis found to be 39° 13' 30".
.'. if cos 8 = 0.77467, (J = 39° 13' 30".
by changing
-
3 = 0 for
Since there is no angle with a cosine equal to -3,
solution admissible is a = cos-1 1 = 0°.
the only
Example
4.-Solve
the equation
cos2 a
+
2 cos a
values of a not greater than 90°.
Solution.-Factoring
the equation,
(cos a
Equating
+ 3) (cos
a-I)
= O.
each factor to 0 and solving for cos a,
.'.
a
=
cos a = 1 and - 3.
cos-1 1 and a = cos-1
(-3).
+
sec2 8 to 1
7 tan2 (J - 4(1
tan2 8, which gives
+ tan2
Simplifying,
(J) + 3 = O.
3 tan2 (J- 1 = O.
Solving for tan (J,
Or
(J = tan-1
10,
tan (J = :!::h/3.
and 8 = tan-1 (-10).
The first of these gives 8 = 30°, which is the only value of 8 less
than 90°.
Solve orally the following
not greater than 90°:
1. sin () = 1.
6.
2. sin () = !V2.
7.
3. sin () = !.
8.
4. cos () = 1.
()
. /= cos-1 !v 2.
EXERCISES
trigonometric
equations
tan () = 1.
()
csc
= 2.
()
tan
= 0.
1
9. cot () =
v:f
10. sec ()
= v'2.
5. cos () = 0.
2
Solve orally the following
angles not greater than 90°:
16.
In using Table IV for finding this value of (J, interpolation is
required.
If the method is not familiar, the explanation will be
found on page 30 of the Tables.
45
This can be checked by substituting 0° for a in the original
equation.
Example 5.-Solve 7 tan2 (J - 4 sec2 (J
+ 3 = 0 for values of (J
not greater than 90°.
Solution.-First
transform so that but a single function is
involved.
This can be done in many ways, but very readily
sin 8 = !, and sin 8 = 1.
The trigonometry part of the solution is to find 8 from these
equations.
They are solved by knowing the values of 8 when
sin 8 = ! and sin 8 = 1. They give
8 = sin-I! = 30°, and 8 = sin-II = 90°.
TRIGONOMETRIC FUNCTIONS
20.
11. sec () =
12. sin () =
()
13. csc
=
14. csc () =
15. csc
anti-trigonometric
a
= tan-1
for values of the angles
equations
1.
o.
V2.
1.
()
= 2..
V3
for values of the
1
24. fJ = csc-1 1.
17. () = sin-1 O.
21. a = tan-1 O.
25. fJ = cot-1 O.
22. a = csc-1 2.
18. () = tan-1 va.
26. l' = cot-1 V3.
23. a = sec-1 2.
19. () = sec-1 V2.
27. l' = csc-1 V2.
Use Table IV in solving the following trigonometric
equations for values
of the angles not greater than 90°:
28. sin () = 0.50628.
33. cot () = 3.6245.
38. cos () = j.
29. cos () = 0.85249.
34. sin () = 0.74896.
39. () = cos-1 !.
30. tan () = 0.58124.
35. cos () = 0.61520.
40. () = tan-1 V2.
31. cot () = 1.6372.
36. cot () = 3.2790.
41. () = cot-1 1.
32. sin () = 0.27148.
37. cos () = 0.57200.
42. () = cos-1 -,s..
Solve the following trigonometric
equations for values of the angles not
greater than 90°:
43. sin2 () sin () = O.
Ans. 0°, 90°.
44. (cos () - 1)(2 cos () - 1)
O.
Ans. 0°, 60°.
0'
-
45. tan4 () - 9 = O.
=
A ns. 60°.
I
l
46
PLANE AND SPHERICAL TRIGONOMETRY
46. sec20
= 4 tan2 O.
0
cot 0) = 4.
tan 0 = 2 cos O.
tan20 - 2V3 tan 0
1 = O.
1) sin 0
V2 =
sin2 0 - 2(V2
V2) cos 0 V2
cos 2 0 - (2
47. V3(tan
48.
49.
60.
61.
62.4
3
3
4
2
+
cos2 0 -
Ans. 300.
Ans. 300, 600.
Ans. 300.
+
2(1
63. 3 tan2 0 - 4V3
+
+
+
V3)
cos
tan 0
+4
+
+
0 + V3
= O.
Ans. 300.
=
O.
O.
Ans. 300, 450.
Ans. 00, 450.
=
O.
Ans. 300, 600.
64. 2 cos 0 - cot 0 = O.
66. 4 sin 2 0 - 5 sin 0 1 = O.
66. tan 0 (sec0 - V2) = V3 (see0 - V2).
+
+
67. 4 sin2 0 - 3V6 sin 0
3 = O.
68. 7 cos2 0 - 29 cos 0 + 4 = O.
69. 2 cos2 0 - sin 20 = O.
60. tan 0
see 0).
4 = 2(sin 0
+
+
61. sin3 0 - cos3 0 = O.
62. 4 tan2 0 = 3 sec2 O.
63. tan 2 0 - 4 tan 0
1 = O.
64. sec 0 - 1 = (V2 - 1) tan O.
+
Ans. 49° 6.4'.
Ans. 30~, 90°.
Ans. 14° 28' 39", 90°.
Ans. 45°, 60°.
Ans. 37° 45.7'.
Ans. 81 ° 47.2'.
Ans. 54° 44' 8".
Ans. 60°.
Ans. 45°.
Ans. 60°.
Ans. 15°, 75°.
Ans. 00, 45°.
CHAPTER
RIGHT
IV
TRIANGLES
37. General statement.-One
of the direct applications of
trigonometry is the solution of triangles both right and oblique.
It is in this way that the surveyor determines heights and distances that cannot be measured directly; for instance, the
height of a mountain or the distance from one point to another
where a lake or a mountain prevents direct measurement.
It
is well to note, however, that the solution of triangles is not the
phase of trigonometry that is of most importance to the student
who is to pursue more advanced subjects in mathematics.
He
will more often find use for the relations existing between the
different functions, and in transforming one form of an expression
involving trigonometric functions into an equivalent one.
It is a recognized fact in all walks of life, and it is certainly
ingrained in mathematical science, that every real advance goes
hand in hand with the invention of sharper tools and simpler
llleLhutb. Practical geometry was developed in Egypt (0 help
redetermine boundaries of the land after an overflow of the Nile.
At an early date astronomy gave the main incentive for the
development of trigonometry.
In attacking the triangle, trigonometry, in many ways, is a
more powerful tool than geometry, which makes little use of the
angles, while trigonometry makes use of the angles, as well as of
the sides, of a triangle.
38. Solution of a triangle.-Every
triangle, whether right or
oblique, has six parts, viz., three sides and three angles. When
certain ones of these are given, the others can be found.
The process of finding the parts not given is called the solution
of the triangle.
By means of trigonometry a triangle can be
solved when the parts given are sufficient to make a definite
geometrical construction
of the triangle.
By geometry, a
triangle can be constructed when three parts are given, at least
one of which is a side. The remaining parts can then be measured and so a solution of the triangle obtained.
47
48
PLANE AND SPHERICAL TRIGONOMETRY
There are two ways of solving a triangle:
(1) The graphical solution.
(2) The solution by computation.
39. The graphical solution.-This
consists in drawing a
triangle such that its angles are equal to the given angles, and
its sides equal to or proportional to the given sides. Of course,
it is necessary that the given parts be consistent and sufficient
to determine a definite triangle.
For instance, two angles must
not be given such that their sum is greater than 180°; nor can a
construction be made if three sides are given such that one of
them is as great as or greater than the sum of the other two.
RIGHT
40. The solution of right triangles by computation.-In
the
two previous articles, what was said referred to the oblique
triangle as well as to the right triangle; here reference is to the
right triangle only.
Since in a right triangle the right angle is always a given part,
it is necessary to have given only two other parts, at least one of
which is a side.
49
In what follows a, b, and c represent the
hypotenuse respectively, and A, B, and C,
the respective sides.
The solutions depend upon the following
two of which are from geometry and the last
nitions of trigonometric functions:
(1) c2
=
a2
+ b2.
(6)
(2) A + B = 90°.
(3)
sin A =
(4)
cos B =
EXERCISES
1. Construct
triangles
by means of the straightedge
and compasses,
having given:
(a) Two sides and the included angle.
(b) Two angles and the included side.
(c) Three sides.
(d) Two sides and an angle opposite one of them.
Discuss and give
drawings for all the possibilities.
(e) Three angles.
Is the construction
definite?
2. Construct right triangles by means of the straightedge and compasses,
having given:
(a) Two legs.
(b) An acute angle and the hypotenuse.
(c) Au aeute angle aud vue leg.
(d) The hypotenuse and a leg.
Use the protractor
in measuring the angles and construct the following:
(a) A right triangle with an acute angle equal to 42° and adjacent side
3.75 in.
(b) An oblique triangle with an angle equal to 35° 16' and the inc~uding
sides 9 and 18 in., respectively.
(c) A triangle with two angles 41° and 63°, respectively,
and the side
opposite the first angle 7.5 in.
(d) A triangle with sides 7.3, 4.5, and 3.8 in., respectively.
(e) A triangle with sides 11.5 and 4.7 ft. and the angle opposite the second
side 120°.
TRIANGLES
c
c
b
cos A = -.
c
(5)
altitude, base, and
the angles opposite
relations, the first
eight from the defi-
sin B
=
~.
c
(7) tan A = ~.
b
(8) cot B = ~.
b
(9) cot A = ~.
a
(10) tan B = ~
a
~.
~.
Number (2) shows that no other part can be derived from the
two acute angles alone. In each of the other formulas, three
parts are involved.
If any two of these parts are given, the third
a
can be found. Thus, in (3) if a and A are given, c = .
sm A i
if c and A are given, a = c sin A i and if a and c are given,
A. = J:5in-1~.
C
.
Exercise.-Solve
each of the above formulas for each letter in
terms of the others.
SOLUTION OF RIGHT TRIANGLE BY NATURAL FlTNCTIONS
41. Steps in the solution.-In
solving a triangle, it is of the
greatest importance to follow some regular order. The following
is suggested:
(1) Construct the triangle carefully to scale, using compasses,
protractor, and ruler. The required parts can then be measured
and a check obtained on the computed values.
(2) State the given and the required parts, and write down the
formulas which are needed in the solution, solving each for the part
required. In choosing these formulas, select for each part
required a formula that shall contain two known parts and
one required part. Thus, if A and a are the given parts and c
50
RIGHT TRIANGLES
PLANE AND SPHERICAL TRIGONOMETRY
51
These agree to four significant figures.
the required part, then sin A = ~ contains the given parts and the
c
a . In general,
required part c. This solved for c gives c = .
sm A
avoid the use of c2
=
a2
+ b2 unless
a table of squares
The formula sin B =
B
a = 3.25.
Given A = 47° 25.6'.
{
b = 2.986.
To find * c = 4.413.
{ B = 42° 34.4'.
Construction
B
.
a
G Iven/ ~
b
c
To find A
jB
tan A = ~
b
sin A = ~
c
A + B = 90°
(3)
b
0.7364
B = 90° - 47° 25.6' = 42° 34.4'.
Check
a2 = c2 - b2 = (c + b)(c - b).
3.252 = (4.413 + 2.986)(4.413 - 2.986).
10.5625 = 10.5584.
when work is completed.
b
c
Formulas
tan A --fj
(2)
cot B --fj
(3)
sin A = ~
c
C
. . . A = tan-1 a
b"
a
.
. . B = cot-l
b"
.
a
. .c =
sin A.
a
Computation
672
A = tan-1
tan-1 2.0550 = 64° 3.1'-.
3:27 =
b = -~_..
tan A
. . c=-. a
sin A
. . . B = 90° - A.I
3.25
2 6
b =
1.0885 = .98.
3.25
c =
= 4 .4 13.
to be inserted
a
a
(1)
B = cot-l
c =
cot-l 2.0550 = 25° 56.9' +.
~:~; =
= 7.473+.
0.~;:19
Computation
* Results
3.27.
7.473.
64° 3.1'.
25° 56.9'.
a
A
(2)
= 6.72.
=
=
=
=
A
Formulas
(1)
also be used in checking.
c
Example 2.-Given a = 6.72 and b = 3.27; find c, A, and B.
Solution.
and square
roots is at hand.
(3) Compute by substituting the given values in the formulas and
evaluating.
(4) Arrange the work neatly and systematically, as this conduces
to accuracy and therefore speed.
(5) Always check. This can be done by making a careful
construction, and also by using other formulas than those used in
the solution.
Example 1.-Given a = 3.25 and A = 47° 25.6'; find b, c, and
B.
Construction
Solution.
~ could
Check
a2 = c2 - b2 = (c + b)(c - b).
6.722 = (7.473 + 3.27)(7.473 - 3.27).
45.158 = 10.743 X 4.203 = 45.153.
It is to be noted that, in computing c, the angle A was used.
Though A was not a given part, it was used to avoid the formula
+
c = Va2
Solve the
formulas
b2.
following
(c + a)(c
-
right
a)
=
EXERCISES
triangles using
b2 and
(c
+ b)(c
natural
-
b)
=
functions.
a2 as a check.
Use the
PLANE AND SPHERICAL TRIGONOMETRY
52
Given
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
b
a
a
b
a
b
c
c
a
a
b
b
c
a
= 32, B = 35°;
;,.
11, A = 43°;
77,
A = 72° 30';
=
= 130, B = 67° 15';
= 27, C = 45;
= 100, A = 70°;
= 30, A = 51°;
= 130, B = 22° 28';
= 40, B = 29°;
= 48, b = 26;
= 150, c = 200;
Find
a, c, A. Check.
b, c, B. Check.
b, c, B. Check.
a, c, A. Check.
b, A, B. Check.
a, c, B. Check.
a, b, B. Check.
a, b, A. Check.
b, c, A. Check.
c, A, B. Check.
a, A, B. Check.
RIGHT
53
TRIANGLES
29. The base of an isosceles triangle is 40 ft. and the vertex angle is 48°
30'; find the equal sides and the base angles.
Ans. 48.7 ft.; 65° 45'.
ao. One side of a regular pentagon inscribed in a circle is 8 in.; find the
radius of the circle.
Ans. 6.8 in.
31. One side of a regular octagon inscribed in a circle is 15 in.; find the
radius of the circle.
Ans. 19.6 in.
= 7.636, B = 73° 45.7'; A = 16° 14.3', a = 2.224, c = 7.9534.
= 0.532, B = 50° 21.9';
= 192.56, b = 437.98;
15. c = 65.8, A = 47° 59.8';
16. b = 1.30, B = 79° 27';
0.4097.
A = 39° 38.1', a = 0.3394, b =
A = 23° 44', B = 66° 16', c = 478.44.
B
= 48.897, b = 44.032.
en
~
= 42° 0.2', a
A = 10° 33', a = 0.242, c = 1.322.
A = 51° 46.4', B = 38° 13.6', a = 6.615.
17. b = 5.21, C = 8.42;
769.96;
A
18. b = 52.02, C =
= 86° 7.6', B = 3° 52.4', a = 768.22.
19. b = 89.49, A = 3° 47.6'; B = 86° 12.4', a = 5.934, c = 89.685.
20. a = 0.1515, A = 40°46.9'; B = 49° 13.1', b = 0.1757, c = 0.232.
21. The shadow of a flagpole 75 ft. high is 98 ft. What is the angle of
elevation of the sun at that instant?
Ans. 37° 25.6'.
22. If side a is three times side b in a right triangle, find angle A.
Ans. A = 71° 33.9'.
23. What angle does a mountain slope make with the horizontal plane
if it rises 450 ft. in 60 rods on
B
the horizontal?
Note: 1 rod =
lot ft.
Ano. 240 26.6'.
24. What is the angle of
inclination of a stairway with
the floor if the steps have a
Run
D tread of 10 in. and a rise of 8
A
C
in.?
Ans. 38° 39.6'.
"""
FIG. 36.
25. What angle does It rafter
make with the horizontal if it has a rise of 6 ft. in a run of 15 ft. ?
Ans. 21° 48.1'.
26. Certain lots in a city are laid out by lines perpendicular to B street,
and running through to A street as shown in Fig. 37. Required the width
of the lots on A street if the angle between the streets is 35° 50'.
Ans. 123.35 ft.
27. Find the angle between the rafters and the horizontal in roofs of the
following pitches: two-thirds, half, third, fourth.
Ans. 53° 7.75'; 45°; 33° 41.4'; 26° 33.9'.
No/e.-By the pitch of a roof is meant the ratio of the rise of the rafters
to twice the run, or, in a V-shaped roof, it is the ratio Df the distance from
the plate to the ridge, to the width of the building.
28. One of the equal sides of an isosceles triangle is 5.74 in. and one of the
base angles is 23° 35'; find the altitude and the base.
Ans. 2.296 in.; 10.521 in.
100'
100'
B St.
100'
100'
I
FIG. 37.
32. One side of a regular decagon inscribed in a circle is 8.56 in.; find the
radius of the circle.
Ans. 13.85 in.
33. One side of a regular octagon circumscribed
about a circle is 12.8 in.;
find the radius of the circle.
Ans. 15.45 in.
34. The radius of a circle is 24 in.; find the side of a regular inscribed
ventag;on.
Of a regular cireulllseribed
pentagon.
.Jns. 28.2 in.; 34.9 in.
Find the areas of the following isosceles triangles:
35. Altitude 28 ft. and base angles each 55° 27'. Ans. 539.85 sq. ft.
36. Base 35.6 ft. and base angles each 64° 51'. Ans. 674.85 sq. ft.
D
90'\....
~0!Hr....
----L...J C
G
FIG. 38.
37. Equal sides each 10.8 in. and vertex angle 48° 17'. Ans. 43.53 sq. in.
38. The radius of a circle is 11 ft. What angle will a chord 14 ft. long
subtend at the center?
Ans. 79° 2.5'.
39. The chord of a circle is 12 ft. long and the angle which it sub tends
at the center is 41.6°. Find the radius of the circle.
Ans. 16.9 ft.
40. Five holes are drawn on a piece of steel with their centers equally
spaced on the circumference of a circle 10 in. in diameter.
Find the distance
in a straight line between the centers of two consecutive holes.
Ans. 5.9 in.
TRIANGLES
RIGHT
PLANE AND SPHERICAL TRIGONOMETRY
54
Logarithmic formulas
41. Thirty holes are drawn with their centers equally spaced on the
circumference
of a circle 22 in. in diameter.
Find the distance between
the centers of two consecutive
holes.
Ans. 2.3 in.
42. Using Fig. 38, with the dimensions as given, find AB.
Ans. 23.61 in.
SOLUTION OF RIGHT TRIANGLE BY LOGARITHMS
42. Remark on logarithms.-By
the use of logarithms, the
processes of multiplication,
division, raising to powers, and
extracting roots may be shortened.
In the solution of triangles,
logarithms are very advantageous in saving time and labor, and
thus conduce to accuracy.
The student should bear in mind,
however, that logarithms are not necessary for this work. The
computer must decide for himself whether or not it will be of
advantage to use logarithms in any given problem.
Formulas which have been so arranged that they involve only
operations of multiplication,
division, raising to powers, and
extracting roots are said to be adapted to computation
by
logarithms.
43. Solution of right triangles by logarithmic functions.- The
solution of a right triangle is the same by logarithms as by natural
functions, except that logarithms are used to avoid the long multiplications and divisions. The tables of logarithmic functions
are used instead of the tables of natural functions.
Example 1.-lhven
a =
b.
Construction
Solution.
(1)
(2)
(3)
log sin A = log a - log c.
log cos B = log a - log c.
log b = log c + log cos A.
Computation
log a = 1.52827
log c
=
1.66011
log sin A = 9.86816 - 10
A = 47° 34.6'
log cos B = 9.86816 - 10
B = 42° 25.4'
Example 2.-Given
and a.
Solution.
b = 8.724 and A = 29° 52.3'; find B, c
Construction
B
To find
B = 60° 7.7'.
10.061.
c::
{ a - 5.011.
a
A
sin A = ~.
c
a
cos B = -.
c
b-, or b = c cos A.
cos A =
c
to be inserted
(1)
(2)
Formulas
* Results
a
A
b
Formulas
A = 47° 34.6'.
42° 25.4'.
To find * B
{ b ::- 30.843.
(3)
a = 33.75
log b = 1.48916
b = 30.843
a = 33.75
.
GIven { c = 45.72.
(2)
Check
a2 = c2 - b2 = (c + b)(c - b)
= 76.563 X 14.877
log (c + b) = 1.88402
log (c - b) = 1.17251
log a2 = 3.05653
log a = 1.52827
log c = 1.66011
log cos A = 9.82905 - 10
B
(1)
55
when work is completed.
b
A
+
B = 90°, or B = 90°
tan A =
c
(3)
cosA
a
TJ'
- A.
or a = b tan A.
b
b
= -, orc =-.
c
cos A
Logarithmic formulas
(1)
(2)
log a = log b
log c = log b
+ log tan
A.
- log cos A.
C
56
PLANE AND SPHERICAL TRIGONOMETRY
Computation
Check
b2 = c2 - a2 = (c + a) (c - a)
= 15.072 X 5.050
log (c + a) = 1.17817
B = 90° - 29° 52.3' = 60° 7.7'
log b = 0.94072
log tan A = 9.75919
-
10
log a = 0.69991
log (c - a) = 0.70329
log b2 = 1.88146
a = 5.0109
log b = 0.94072
log cos A = 9.93809
log c
-
log b = 0.94073
b = 8.7242
10
= 1.00263
c = 10.061
No/e.-It
is best to make a full skeleton solution before proceeding to the
use of the Tables.
The skeleton solution can be seen in this example by
erasing the numerical quantities.
x
In using the Tables, plan so as to save time as much as possible.
For instance, if both log sine and log cosi
required, look up both of them while the tables are open at that
page.
1'1- ~
''1- i\
'd. ' ~
<A",.
'
.
",.
.~
'"'~6
w
€.bys.
'€s,,g
.
s.~
<1'~ '6Jo~
"'~<!,'
",\P1'd.
'.,. \P
';. l' '"<I'
<tj
.
FIG. 40.
~."'''=''-'
O¥ ",'.s
"'
s
44. Definitions.-The
angle of elevation is the angle between
the line of sight and the horizontal plane through the eye when
TRIANGLES
57
the object observed is above that horizontal plane. When the
object observed is below this horizontal plane, the angle is called
the angle of depression.
Thus, in Fig. 39a, an object 0 is seen from the point P. The
angle 8 between the line PO and the horizontal PX is the angle
of elevation.
In Fig. 39b an object 0 is seen from the point of
observation P. The angle 8 between the line PO and the horizontal AP is called the angle of depression.
Directions on the surface of the earth are often given by
directions as located on the mariner's compass.
As seen from
Fig. 40, these directions are located with reference to the four
cardinal points, north, south, east, and
west. The directions are often spoken of
as bearings.
Present practice, however,
B
gives the bearing of a line in degrees. The
E
bearing of a line is defined to be the acute wangle the line makes with the north-andsouth line.
c
Thus, in Fig. 41, if 0 is the point of
Is
FIG.41.
observation, the bearing of OA is north, 20°
east, written N. 20° E.
that of OC is S. 30° E.
FIG. 39.
""'ft.
"'ItIJ:N. "
'"
\ll:b~"'.'j
RIGHT
The bearing of OB is N. 60° W., and
EXERCISES
Solve the following
right triangleR for the partR not given.
The firRt two
1. a = 31.756, A = 54° 43.5'. Check results.
2. b= 13.98, B = 21 ° 54'. Check results.
3. b = 1676.34, c = 5432.8. Check results.
4. a = 4.5612, B = 43° 3.7'. Check result£..
5. b = 54.78, A = 35° 43.2'. Check results.
6. a = 25.13, c = 43.412. Check results.
'1. c = 23.746, A = 32° 54.21'. Check results.
8. a = 134.90, b = 101.43. Check results.
9. a = 14.23, b = 9.499. Check results.
10. c = 143.89, B = 39° 54.8'. Check results.
11. a = 18.091, b = 1378.2. Check results.
12. a = 896, B = 2° 6' 10". Check results.
13. a = 653, c = 680, b = 189.7, A = 73° 48', B = 16° 12°.
14. b = 675.31, B = 78° 34.6', a = 136.46, c = 688.97, A = 11° 25.4'.
15. b = 1100, c = 1650, a = 1229.9, A = 48° 11.4', B = 41 ° 48.6'.
16. c = 11.003, A = 45° 32' 19", a = 7.8530, b = 7.7067, B = 44° 27.7'.
1'1. a = 0.001348, b = 0.0009896, c = 0.0016722, A = 53° 43', B = 36°
17'.
18. A ladder 30 ft. long rests against a building standing on level ground,
and makes an angle of 65° 35' with the ground. Find the distance it reaches
up the building.
Ans. 27.3 ft.
58
PLANE
AND
SPHERICAL
TRIGONOMETRY
19. A tower stands on level ground.
At a point 161.7 ft. distant and
5.5 ft. above the ground the angle of elevation of the top of the tower is
62° 48'. Find the height of the tower to the nearest foot.
Ans. 320 ft.
20. From the top of a tower 375 ft. high, the angle of depression of a man
on the horizontal plane through the foot of the tower is 37° 24.6'.
Find the
distance the man is from the foo~ of the tower.
Ans. 490.3 ft.
21. What is the angle of inclination of a roadbed having a grade of 14 per
cent?
One with a grade of 26 per cent?
(A road with a rise of 14 ft. in
100 ft. on the horizontal has a grade of 14 per cent).
Ans. 7° 58.2'; 14° 34.4'.
22. Locate the centers of the holes Band
C (Fig. 42) by finding the
distance each is to the right and above the center O. The radius of the
circle is 4.5 in. Compute correct to four decimals.
Ans. (3.6406, 2.6450); (1.3906, 4.2798).
D
F
A
FIG. 42.
/j t/
b
FIG. 43.
23. In the parallelogram of Fig. 43, b = 33.7 in., c = 14.8 in., and
(j
=
126° 15'. Find the altitude h of the parallelogram.
Ans. 11.94 in.
24. A ladder 32 ft. long is resting again~t a wall at an anp-k of ~1. 7°.
Ifthe-£o&t~f
tMlaooef" is tll'aWft-away-4ft;,-how
faT down thewntlwitlttm
top of the ladder faIl?
Ans. 1.9 ft.
25. A man surveying a mine, measures a length AB
1240
ft. due east
=
with a dip of 6° 15'; then a length BC = 3425 ft. due south with a dip of
10° 45'. How much deeper is C than A?
..lns. 773.84 ft.
26. Find the number of square yards of cloth in a conical tent with a
circular base, and vertical angle 78°, the center pole being 12 ft. high.
Ans. 52.4 sq. yd.
Find the areas of the following isosceles triangles:
27. Altitude is 27 ft. and base angles each 55.6°. Ans. 499.2 sq. ft.
28. Base is 3 ft. and vertical angle 38° 24'.
Ans. 6.46 sq. ft.
29. Each leg is 15 ft. and base angles cach 63° 18.6'. Ans. 90.29 sq. ft.
30. Find the area of a regular pentagon one of whose sides is 10 in.
Ans. 172.05 sq. in.
31. Find the area of a regular octagon one of whose sides is 15 in.
Ans. 1086.4 sq. in.
32. Find the difference in the areas of a regular hexagon and a regular
octagon, each of perimeter 80 ft.
Ans. 20.96. sq. ft.
33. Prove that the area of a right triangle is given by each of the following, where S is the area:
RIGHT
TRIANGLES
59
S = !bc sin A.
S = !ac cos A.
S = !C2sin A cos A.
34. If R is the radius of a circle, show that the area of a regular circumscribed polygon of n sides is given by the formula:
A = nR2 tan 180° .
n
35. Show that the area of a regular inscribed
the formula:
A
=
.
nR2 sm
-180°n
cos
-180°n
=
polygon of n sides is given by
1
-nR2
2
. -.360°
SIn
n
36. The radius of a circle is 30 in. Find the perimeter
of a regular
inscribed pentagon.
Ans. 176.34 in.
37. Find the area of a regular octagon inscribed in a circle whose radius is
8 in.
Ans. 181.02 sq. in.
38. What diameter of stock must be chosen so that a hexagonal end 3 in.
across the flats may be milled upon it?
Ans. 3.46 in.
Answer the question for an octagon.
The meaning of "across the flats" is shown in Fig. 44.
Ans. 3.25 in.
39. From a point 460 ft. above the level of a lake
the angle of depression of a point on the near shore
is 21 ° 56', and of a point directly beyond on the
opposite shore is 4° 31'. Find the width of the lake.
FIG. 44.
Ans. 4680.7 ft.
40:-Fln:G the angles at the base made by-the sides ofa towel' witl1the
horizontal, if the tower is 47 ft. 6 in. high, has a square base 6 ft. on a side,
and a top 8 in. square.
Ans. 86° 47.2'.
41. Suppose the earth a sphere with a radius of 3960 miles; find the length
of the arctic circle which is at latitude 66° 32'.
Ans. 9908.3 miles.
42. Find the length of 1 ° of longitude in the latitude of Chicago, 41 ° 50',
if the earth is a sphere with a radius of 3960 miles.
Ans. 51.497 miles.
43. A circle 12 in. in diameter is suspended from a point and held in a
horizontal position by 12 strings each 8 in. long and equally spaced around
the circumference.
Find the angle between two consecutive
strings.
Ans. 22° 23.2'.
44. A girder to carry a bridge is in the form of a circular arc.
The length
of the span is 120 ft. and the height of the arch is 30 ft. Find the angle at
the center of the circle such that its sides intercept the arc of the girder;
and find the radius of the circle.
Ans. 106° 15.7'; 75 ft.
45. A tree stands upon the same plane as a house whose height is 65 ft.
The angle of elevation and depression of the top and base of the tree from
the top of the house are 45° and 62°, respectively.
Find the height of the
tree.
Ans. 99.6 ft.
46. From a point 20 ft. above the surface of the water, the angle of elevation of a tree standing at the edge of the water is 410 15', while the angle of
l
60
RIGHT TRIANGLES
AND SPHERICAL TRIGONOMETRY
PLANE
depression of its image in the water is 58° 45'. Find the height
and its horizontal distance from the point of observation.
of the tree.
Ans. 51.88 ft.; 65.50 ft.
47. The legs supporting a tank tower are 50 ft. long and 18 ft. apart at
the base, forming a square.
The angle which the legs make with the
horizontal
line between the feet diagonally opposite is 83° 30'. How far
apart are the tops of the legs?
48. The angle of elevation of a balloon from a point due southAns.
of it10is ft.50°,
and from another point 1 mile due west of the former the angle of elevation
is 40°. Find the height of the balloon.
Ans.
1.18 miles.that
49. At a point P on a level plain the angle of elevation of
an airplane
is southwest of P is 38° 35'. At a point Q, 2 miles due south of P, the airplane
appears in the northwest.
What is the height of the airplane?
61
55. A wall extending east and west is 8 ft. high.
The sun has an inclination of 49° 30' and is 47° 15' 30" west of south.
Find the width of the
shadow of the wall.
Ans. 4.637 ft.
56. A tripod is made of three sticks, each 5 ft. long, by tying together the
ends of the sticks, the other ends resting on the ground 3 ft. apart.
Find
the height of the tripod.
Ans. 4.690 ft.
57. At a certain point the angle of elevation of a mountain peak is 40°
30'. At a distance of 3 miles farther away in the same horizontal plane, its
angle of elevation is 27° 40'. Find the distance of the top of the mountain
above the horizontal plane, and the horizontal distance from the first point
of observation to the point directly below the peak.
An.~. 4.77 miles.
B
Ans. 1.13 miles.
Suggestion.-Find
h and. x representing
30'
E
F
FIG. 45.
50. From a point on a level plain the angle of elevation of the top of a hill
is 23° 46'; and a tower 45 ft. high standing on the top of the hill subtends an
angle of 5° 16'. Find the height of the hill above the plain.
Ans. 172.7 ft.
51. A flagstaff stands upon the top of a building 150 ft. high.
At a
horizontal
distance of 225 ft. from the base of the building the flagstaff
subtends an angle of
6° 30'. Find the height of the flagstaff.
40.07 ft.
52. Two observers are stationed 1 mile apart on a straight Ans.
east-and-west
level road.
An airplane flying north passes between them, and, as it is
over the road, the angles of elevation are observed to be 72° 30' and 65° 15'.
Find the height of the airplane.
1.29 miles. is
53. A ship is sailing due east at 16 miles per hour. Ans.
A lighthouse
observed due south at 8:30 A.M. At 9:45 A.M. the bearing of the same lighthouse is S. 38° 30' W. Find the distance the ship is from the lighthouse at
the time of the first observation.
25.14
54. Find the width of the shadow of the wall shown Ans.
in Fig.
45. miles.
If the
height of the wall is h ft., the angle of elevation of the sun a, and the angle
between the vertical plane through the sun and the plane of the wall 8, show
that width of shadow
h cot a sin 8.
=
two simultaneous
equations involving the unknowns
the distances as shown in Fig. 46. These are tan 40°
Solve these algebraically
for h. and x.
and tan 27° 40' =
3 ~ x'
~
58. At a certain point A the angle of elevation of a mountain peak is a;
=
at -&~t
E thatisa miles f&J:theraway-in
-the-SaIIill
horizontal
plane its
angle of clevatlOII l~ p. II It represents the distance 1hf' peak is above (,he
plane and x the horizontal distance the peak is from A, derive the formulas:
atanatanp.x
h=
tan a - tan 13'
atanp
=
tan a - tan 13
Note.-In
using these formulas, it is convenient to use natural functions.
In Exercise 5, page 150, is given a solution of the same problem, obtaining
formula,s adapted to logarithms.
59. Find the height of a tree if the angle of elevation of its top changes
from 35° to 61 ° 30' on walking toward it 200 ft. in a horizontal line through
its base.
Ans. 225.93 ft.
60. A man walking on a level plain tOward a tower observes that at a
certain point the angle of elevation of the top of the tower is 30°, and, on
walking 305 ft. directly toward the tower, the angle of elevation of the top
is 52°. Find the height of the tower if the point of observation each time
is 5 ft. above the ground.
Ans. 325.8 ft.
61. At a certain point the angle of elevation of the top of a mountain is
36° 15'. At a second point 700 ft. farther away in the same horizontal plane
the angle of elevation is 28° 30'. Find the height of the mountain above
the horizontal plane.
Ans. 1464.6 ft.
FUNCTIONS
OF LARGE ANGLES
63
see (!1r - 8) = r' = r = csc 8.
?
CHAPTER
FUNCTIONS
csc (!1r
V
OF LARGE ANGLES
45. It is proved in Art. 16 that, for any angle, each of the
trigonometric functions has just one value. On the other hand,
it was shown later that a particular value of a function may go
with more than one angle. For instance, sin-I! is 30° and 150°
and, in fact, may be anyone of the other angles whose terminal
sides lie in the same positions as the terminal side of 30° or
150°. This would suggest that possibly any function of a large
angle may be equal to a function of an angle that is not greater
than 90°. Further, it would seem that some such relation must
exist for the tables have only the functions of angles of 90° or
less tabulated.
We shall now proceed to
y
p'
show that a function of a large angle can be
p expressed as a function of an angle less than
90°.
y
46. Functions of!~ - 6 in terms offunctions of 6.-It has been shown in previous
articles that a_function of an acute angle is
:.;f..,tctitid of tlie nmlpicmcnt of
FlU. 47.
that angle. This will now be proved
in a different manner.
Let 8 be any acute angle drawn as in Fig. 47. Construct
-!1r- 8, take OP' = OP, and let x, y, and r be the abscissa,
ordinate, and distance, respectively, of Pi and x', y', and r' those.
for P'. It is evident that right triangles OMP and 0' M'P' are
equal.
Then, since y' = x, x' = y, and r' = r,
,
-
8) =
~ = ~ = cos 8.
(!1r -
? = ~ = sin 8.
sin (!1r
f
cos (!1r
8)
tan
y'
x
8) = -,
x = -Y = cot 8.
=
cot (-!1r- 8) =
?
62
=
~
=
tan 8.
-
8)
Y
r'
r
= 1/ = x = sec 8.
Notice that in each line the function at the end is the cofunction
of the one at the beginning.
47. Functions of!~ + 6 in terms of functions of 6.-In Fig. 48,
let 8 be any acute angle. Construct !1r + 8, take OP' = OP, and
represent the other parts as shown.
y
p~
y
p:
FIG. 48.
p
FIG. 49.
Then, since y' = x, x' = -y, and r' = r,
,
sin (!1r + 8) = '!!, = :£ = cas 8.
r
r
J.
x'
-y
r'
y'
r
x
+ 8) = - =
tan (1m
2
cot (1.
2 1r
+
see (!1r +
csc (p
+
-
_7J
~
-'~;
r
x
--y = -- Y -cot 8.
x'
8) = -x' = --y = -- Y = -tan (J.
y'
x
x
r'
r
8) =
= y = - yr -csc 8.
:?
-
8) = -,r' = -r = see (J.
y
x
Notice that here, also, in each line the function at the end is the
cofunction of the one at the beginning.
Examples.-sin
130° = sin (90°
cot 110° = cot (90°
+ 40°) = cas 40°.
+ 20°) = -tan 20°.
48. Functions of ~ - 6 in terms of functions of 6.-In Fig. 49,
let 8 be an acute angle. Construct
represent the other parts as shown.
1r
-
8, take op' = OP, and
64
PLANE
Then, since x'
AND
SPHERICAL
TRIGONOMETRY
FUNCTIONS
= -x, y' = y, and 1" = 1',
,
sin ('II"- B) =
~=
l'
l'
cas ('II"- B)
x'
-
- X
tan ('II"-
B)
=
1"
11
=
~
= sin
l'
y'
Y
= x'
-- = -x
B.
= --
Y
B)
1"
=- =
x'
csc ('II"-
B)
l'
~
= -tan
B.
(180°
(180°
B.
-
-
20°)
40°)
=
=
+ a in terms of functions of a.-In
angle. Construct'll" + B, take OP' =
let B be an acute
represent the other parts as shown.
p
('II"
+ B) =
1"
= -
11
l'
l'
l'
y =
230° = tan (180°
cas 205° = cas (180°
sin (.3.
- B)
2 '11"
=
y'
-x
- = 1"
-y
=
- csc
+ 50°)
+ 25°)
B.
= tan 50°.
= - cas 25°.
x
= -- l' = -cas
l'
-
1"
Fig. 50,
OP, and
-
tan (V
B)
=
y'
"Xi
=
y
= -- l'
l'
x
y=
cot
B.
= -sin
B.
-
B.
B.
x
cot (t'll" - B) = x' = Y = tan B.
sec
(~'II" -
B)
=
csc (V - B) =
x
1"
"Xi
r'
11
l'
l'
r
r
= - y = - Jj
csc
= =x = -x = -sec B.
Notice that here again in each line the function at the end is the
cofunction of the one at the beginning.
Examples.-sin
250° = sin (270° - 20°) = -cas 20°.
tan 210° = tan (270° - 60°) = cot 60°.
FIG. 50.
since x'
csc
l'
11
Y
Then,
= --x = -- x = - sec B.
-y
cas (2a'll"- B) = x'
- =
-cas 20°.
csc 40°.
49. Functions of ~
Y
1"
50. Functions of t~ - a in terms of functions of a.-In Fig. 51,
let B be an acute angle. Construct t'll" - B, take OP' = OP, and
represent the other parts as shown.
Then, since y' = -x, x' = -v, and 1" = 1',
l'
-x = -- x = -sec B.
= -y'1" = -Yl' = csc
= cas
= csc
+ B) = xI
Examples.-tan
Notice that in each line the function at the end is the same
function as the one at the beginning.
Examples.-cos
160°
csc 140°
('II"
Notice that here, also, in each line the function at the end is the
same function as the one at the beginning.
X
x'
-x
x
cot ('II"- B) = - = y'
Y = -- Y = -cot B.
sec ('II"-
sec
X
= -- l' = -cas B.
65
OF LARGE ANGLES
FIG. 51.
= -x, y' = -v, and 1" = 1',
.
y'
-y
sm ('II"+ B) = - = 1"
l'
y
.
= - -l' = - sm
51. Functions of t~
let B be an acute
B.
x'
-x - _::
cas ('II" + B)
=r'=rl' = -cos B.
-y
y'
11
tan ('II"+ B) = "Xi--x = X = tan B.
cot ('II"+ B) = x' - -=-:: = ::
- -y
y = cot B.
11
angle.
+ a in terms of functions of a.-In
Construct
t'll" + B, take OP' =
represent the other parts as shown.
Then, since y' = -x, x' = y, and 1" = 1',
B)
sin (JL...
+
cas
+ B)
2"
(t'll"
y'
-x
= -1" = -.=
l'
=
x'
I
l'
y'
=
11
l'
-x
tan (t'll" + B) = '-; = x
y
x
-- l' = -cas B.
= sin B.
x
= - - = - cot B.
Y
Fig. 52,
OP, and
66
PLANE AND SPHERICAL
+
y
x'
Y
= y'- = -x = --x = -tan e.
e) = Ir' = -r = csc 8.
x
y
r'
e) = - = - r = -- r = -see e.
y'
-x
x
-
8)
oot (&71'"
"2"
see (t7l'"
+
+
csc (871'"
"2"
FUNCTIONS
TRIGONOMETRY
Notice that here, also, in each line the function at the end is the
cofunction of the one at the beginning.
Examples.-cot
310°
=
cot (270°
+
40°) = -tan
40°.
sec 340° = sec (270° + 70°) = csc 70°.
52. Functions of - 9 or 2-:; - 9 in terms of functions of 9.In Fig. 53, let 8be an acute angle. Construct 271'"
- e,oP' = OP,
and represent the other parts as shown.
y
y
p
Then, since x' = x, y' = -y, and r' = r,
,
-y
'!L
)
(
sin -e =
=
= _¥.. = -sin 8.
=
tan (- 8) =
cot (-e)
r'
r
r
x'
x
-r' = -r = cos 8.
,
-y
= - tan 8.
= x = - ¥..
x'
x
'!L
x'
-x
x
= -y' = -y = -- y = -cot 8.
sec ( -e ) = -r' =
x'
csc (-8) = r' =
17
r
- = sec 8.
x
r
r
-y = -y = -csc e.
These formulas can readily be remembered by noting that
the functions of the negative angle are the same as those of the
positive angle, but of opposite sign, except the cosine and the
secant, which are of the same sign.
tan (
1r
e)
e)
8)
8)
=
=
=
=
cot
tan
csc
sec
tan
cot
e.
e.
e.
e.
e)
e)
e)
e)
e)
e)
= -sec e.
= csc e.
+ e) =
tan (
1r
cot (
1r
sin (!71'"
=
=
=
=
cot e.
tan 8.
-ese 8.
-see 8.
=
=
=
=
=
=
-sin e.
cos e.
-tan e.
-cot e.
sec e.
- cse e.
cot
(!71'"
see (t7l'"
cos
e.
e) = -see e.
cot 8.
e) = -csc
e.
+ e) = - eos e.
+ 8) = sin e.
+ e) = -cot e.
+
e) = -tan
+ 8) =
+
esc (-~71'"
sin
cos
tan
cot
see
csc
-
+
+ e) =
+
cos (t7l'"
tan (t7l'"
-csc 8.
see e.
-sin e.
e) = tan e.
( 71'"
8.
e.
-sin e.
+
sec ( 71'"
ese
cos
e) = - cot e.
e) = -tan e.
cos (
= -sin e.
= -eos
+
+
+ e) =
1r + e) =
1r + e) =
sin (
8.
- e) = -tan e.
- 8) = - cot 8.
(271'"
(271'"(271'"
(271'"
(21r (271'"
-
(!71'"
(!71'"
sec (!71'"
csc (!71'"
e) = sin e.
cot ( 1r
sec ( 71'"
- e)
cse ( 71'"
- e)
sin (t7l'"- e)
eos (t7l'"- e)
tan (t7l'"- e)
cot (t7l'"- e)
see (t7l'"- e)
ese (!71'"- e)
sin
eos
tan
cot
see
cse
sin (!71'"
cos (!71'"
e) = sin e.
cos ( 71'"- e) = -cos
FIG. 52.
cos ( -8 )
(!1r (!71'"
(!1r (!71'"
(~ ( 1r -
p'
p'
FIG. 53.
67
+ e) =
+ e) =
sin (!71'"
- e) = cos e.
y'
x
ANGLES
53. Functions of an angle greater than 2-:;.-Any angle a
greater than 271'"
has the same trigonometric functions as a minus
an integral multiple of 271'",
because a and a - 2n7l'"have the same
initial and terminal sides. That is, the functions of a equal
the same functions of a - 2n7l'",where n is an integer.
That is, a function of an angle that is larger than 360° can be
found by dividing the angle by 360° and finding the required
function of the remainder.
54. Summary of the reduction formulas.-The
formulas of
the previous articles are here collected so as to be convenient
for reference.
It will be well to memorize the last group, the one
expressing the functions of negative angles as functions of positive angles.
cos
tan
cot
sec
csc
sin
p
OF LARGE
(-8)
(- e)
(-e)
(-e)
(- e)
(-e)
e) =
=
=
=
=
=
=
cse 8.
-see
8.
e.
-sin e.
eos 8.
-tan e.
-cot e.
see e.
-csc 8.
While the proofs of these formulas have all been based upon
the assumption that e is an acute angle, they are true for all
68
PLANE
AND
SPHERICAL
TRIGONOMETRY
FUNCTIONS
{) in
"alues of 0, and can be carried through for any value of
exactly the same manner as for 0 an acut e angle.
Tables of trigonometric functions, in general, do not contain
angles greater than 90°. Since the principal application of the
reduction formulas is made in determining the numerical values
of functions of angles greater than 90°, it will be found convenient
to have a rule for the application of the formulas for 0 an acute
angle. The rule gives a final summary of the preceding articles.
RULE.-Express the given angle in the form n . 90° ::J::0, where
0 is acute. If n is even, take the same function of 0 as of the given
angle; if n is odd, take the cofunction of o. In either case the final
sign is determined by the function of the given angle and the quadrant
in which that angle lies.
If the given angle is negative, first express its function as the
function of the given angle with its sign changed, and then proceed
as before.
Example I.-Find
cos 825°.
Solution.-By
the rule and Table V,
cos 825°
=
cos (9 X 90°
+
15°)
=
-sin
15° = -0.25882.
Since 9 X 90° is an odd number times 90°, we take the sine
of 15°. It is negative because 825° lies in the second quadrant
in which cosine is negative.
Another solution of this is as follows:
cos
-
cos ~~ X iSoU- -t- 1U5-)
+ 15°) = -sin
= cos (90°
=
cos
15°.
Solution.-First
evaluate
each of the functions.
see 371" = see (6 X!7I"
sin 171" = sin (9 X !71"
+ 0)
+
= -see 0 = -1.
0) = cos 0 = 1.
cos 171"= cos (7 X P + 0) = sin 0 = o.
csc J-f7r = csc (15 X !71"+ 0) = -see 0 = -l.
cos -t7l"= cos (5 X !71"+ 0) = sin 0 = O.
see 771"= see (14 X ~ + 0) = -see 0 = -1.
3
CSC
¥71"
+
t7l"
2 cos 171" 2( -1)
7 COS t7l" - see 771" 3( -1)
3 sin 171"
+
-5
=-2=
.
771"
.
Example 4.-Evaluate
sm 0
- 3 . 1
+
+
2
.0
7 . 0 - (-1)
5
2'
6
.
J ~..
Solution.-The
notation given is a form frequently used, and
means that: (1) the upper number 6 is to be substituted for 0;
(2) the lower number i7l"is to be substituted for 0; and (3) the
result of (2) is to be subtracted from that of (1).
6
Then
sin O ~.. = sin 6
J
Since 6 is a number
of radians
-
sin
~71".
and
6 radians = 6(57° 17' 44.8") = 343° 46' 29",
sin 6 = sin 343° 46' 29" = - cos 73° 46' 29" = - 0.27941.
6
.
.'. sm 0J i.. = -0.27941 - (-1) = 0.72059.
EXERCISES
In Exercises 1 to 47 do the work orally.
Express each of the following as
a function of e:
7. tan (720° - e).
1. sin (720° + e).
e).
e).
5. sec (1080° + e).
6. cos (990° + e).
= - cot 35° = -1.4281.
2 see 371"- 3 sin -t7l"+ 2 cos
3 csc l,f~7I"
+ 7 cos -t7l" - see
2 see 371" -
69
ANGLES
these values,
3. sin (630° +
4. cot (630° -
Example 2.-Find
cot (-1115°).
Solution.-First
express as a positive angle and then apply the
rule.
cot (-1115°) = -cot 1115° = -cot (12 X 90° + 35°)
.
E xamp le 3 .- F m d th e va Iue 0f
Substituting
OF LARGE
9. cos
(e - 1080°).
10. tan (e - 810°).
11. csc
(1890°
+ e).
12. sec (2880° - e).
Express the following functions as functions of acute angles.
Give two
answers, one where the angle is less than 45° and one where it is greater.
13. sin 150°.
18. cos (-45°).
23. sin 127° 30'.
14. cos 100°.
19. tan 290°.
24. cos 281 ° 30'.
15. tan 210°.
20. cot 185°.
25. cot 235° 15'.
16. cot 265°.
21. sec 275°.
26. tan 347° 20'.
17. cos 320°.
22. sin (-85°).
27. tan (-68° 30').
Express the following functions as functions of angles less than 45°.
28. sec 165°.
33. cos 2000°.
38. sec (-300).
29. cot 430°.
34. sec 600°.
39. cot (-425).
30. tan 305°.
35. cot 1050°.
40. tan ( -600).
31. cos 195°.
36. csc 840°.
41. sin (-450).
32. sin 145°.
37. sin 700°.
42. cos (-325).
What is the value of each of the following:
43. 3 sin (90° + e) + 4 cos (180°
- e).
44. 3 sin (360° - e) - 3 cos (270° + e).
FUNCTIONS
70
OF LARGE
ANGLES
71
PLANE AND SPHERICAL TRIGONOMETRY
46. 2 tan (180° - (J) - 2 cot (90° + (J).
46. 5 sin (270° + (J) - 3 sin (270° - (J).
47. 4 cos (180° - (J) + 5 sin (270°
+
82. If tan 200° = c, find
0).
Show that the following are true equalities:
48. tan (225°
49.
sin
(135°
-
0)
+
0)
= tan
(45°
= cos (45°
-
+
0).
0).
50. cot (135° + 0) = - cot (45° - 0).
61. tan (45 :t 0) = cot (45 =+=0).
By the use of the table of natural functions, find the sine, cosine, tangent,
and cotangent of the following angles:
62. 156°.
66. 835° 40'.
60. -481°.
63. 215°.
57. 460° 18'.
61. -1301°.
64.268°.
58.934° 52'.
62. -152° 13'.
66. 297°.
69. 1045° 25'.
63. -209° 24'.
64. Find the sine, cosine, tangent, and cotangent
of 135°, 150°, 240°,
330°, 315°, 120°, 210° by expressing them in terms of functions of 30°, 45°,
or 60°. Compare the results with the table of values given on page 24.
sin (!11"- 0)
sin (!11" - 0) cos (!11" + 0)
.
A ns. - 1 .
65 S'Imp l'f1 y
- sec (11"+ 0)
tan (!11" + 0)
Verify Exercises 66 to 71.
.
66.
67.
tan
1
11"-
+ tan
cos!1I"
tan
11"
0
=
tan 0
COS 0 -
tan (11"- 0).
!11" sin
sin
0 =
COS (!11"
+
0).
68. sin %11"
cos 0 - cos l.. sin 0 = sin (!11"- 0).
69. sin (!11"+ a) cos (11"- a) + COS (!11" + a) sin (11"- a) = -1.
sin (90° + 0) + cos (270. - 0) .
sin (-0) + cos (-0)
70 '.
.
= cot (180° + 0) + tan (360° - 0)
.."
(-0)
?T. + ros 3T.)
.
-
. sec
~--Tcsct1l"
--
511"
3
-
2'
Ans. 611".
72. Evaluate 12(!0 - 1 sin 20) ]~.
73. Evaluate
+ cos
(tan x
74. Evaluate
(ix - j sin2x)
76. Evaluate
!a2(!0
+
Ans. -2.
x) J~.
2sinO
2" .
Ans. 2.323.
J t..
+
lsin20)
76. Evaluate 1I"a'(0- 4sinO+ Isin20
.
77. Evaluate
-
cos x
78. Evaluate
-
sin
79. If sin 0
= -H,
Ans.
J:".
+ isin'O) J~.
Am. a'1I"2.
4
Ans. 0.9302.
J: + cos x J ..'
3
x i" + sin x
.
Jh
J1
with 0 in the fourth
2.534a2.
quadrant,
show that vers (0 - 11")
= \\".
80. If cot 250° = ~, show that tan 160° = -~, and sec 430° = Vl+b'.
81 . If covers 115 ° -- I
-! e find
vers 205° cos 335°.
. -. --
v0-=1.
A ns.
- cos 250°.
Ans.
C
C2 + (
sin 250° + tan 290.
83. If csc 160° = c, find
Ans. -1.
cot 200° + cos 340°'
Draw the figures and derive the formulas in each of the following:
84. Functions of 90° + 0 in terms of functions of 0 when 0 is in the third
quadrant.
86. Functions of 270° - 0 in terms of functions of 0 when 0 is in the second
quadrant.
86. Functions of 180° + 0 in terms of functions of 0 when 0 is in the fourth
quadrant.
55. Solution of trigonometric
equations.~All
the angles less
than 360° that have the same absolute value for each of the
trigonometric
functions are called corresponding
angles. In
general, there are four such angles for each trigonometric
function.
For instance, if sin () is ! in absolute value, that is, if
()
sin = I!, then () = 30, 150,210, and 330°. These four angles
are called the corresponding angles when the absolute value of
sin () is
~.
In general, the corresponding angles lie one in each quadrant,
and have their terminal sides placed equally above and below the
x-axis. The exception is when the angles lie between the
quadrants, and then there are but two corresponding angles.
Thus, if sin () = II, the corresponding angles are 90 and 270°.
It follows that, if cpis the angle lying in the first quadrant, then
Ihe of her corresponding angles are. - -If the value of a trigonometric function is given, the angle can
be found by the following:
RULE.-First
find the acute angle cp by the table of natural
functions, using the absolute value 'Jf the given function.
The
remaining, or corresponding, angles which have the same trigonometric function in absolute value are 180° I cp and 360° - cpo
From these four angles the angles can be chosen in the proper quadrants to satisfy the given function.
That is, if the function is positive, the angle is taken in those
quadrants in which that function is positive.
()
= -!; find < 360°.
Solution.-First
find cp = sin-I! = 30°.
By the rule, the remaining angles which have their sine equal
to ! in absolute ~alue are 180° - 30° = 150°,
Example
Am. -0.4313.
:: ~~g:
and
I.-Given
sin ()
180°
360°
+
30° = 210°,
- 30° = 330°.
72
PLANE
AND
SPHERICAL
Since the sine is negative, 8 must be in the third and fourth
quadrants.
. '. 8 = 210 and 330°.
Example
2.-Given
cos 8
= -!V2;
find 8 < 360°.
q, = cos-1 !V2 = 45°.
Solution.-Find
The corresponding angles are 135, 225, and 315°.
But the cosine is negative in the second and third quadrants,
.'. 8 = 135 and 225°.
Example
3.-Given
2 sin 8
+
cos 8 = 2; solve for 8 < 360°.
Solution.-First
express all the functions in terms of one function as in Art. 33. Then, since cos 8 = VI - sin2 8, we have
2 sin 8
Transposing
+ VI
and squaring,
- sin2 8
= 2.
4 sin2 8 - 8 sin 8 + 4 = 1 - sin2 8.
8 sin 8
3 = 0, which is a quadratic
Transposing, 5 sin2 8 +
equation in sin 8.
Solving for sin 8 by the formula,
.
sm 8 =
8:J:
V64
FUNCTIONS OF LARGE ANGLES
TRIGONOMETRY
-
and the angle 8 must be in the third or fourth quadrant.
necessary, then, to reject tll" and ~.
. . . 8 = fIr and tll".
Example 5.-Given
Example 6.-Given tan 28
Solution.-tan
28 = 0.
Then
60
10
3
= 1 or 5'
tan 8 sec 8 = - V2; solve for 8 < 211"
sec 8 = VI + tan2 8,
Solution.-Substituting
Squaring, tan2 8 (1
tan48
Solving,
tan2 8
+ tan2 8)
+ tan2
= -V2.
= 2.
=
=
.'. 8 = tan-1 (:J: 1) =
solve for 8 < 360°.
28 = 60°, 240°, 420°, 600°.
. '. 8 = 30°, 120°, 210°, 300°.
Notice that, in order to find all values of 8 < 360°, we take all
values of 28 < 720°.
Give orally the values of the following
1. sin () =
1 or -2
:J:1 or :J:V="2.
tll", ~,
~11", tll".
Since V -2 is imaginary, no such angle as tan-1 (:J: V -2)
exists.
From the original equation the product of tan 8 and sec 8 is
negative.
Therefore, these functions must be opposite in sign,
()
= !V2.
cos
.
6. sin() =
-t.
()
= COS-l(-tya).
()
12.
()
= cot-l (h/3).
~
HIll
y2
()
= sin-1
l,an
()
9. sin = tV3.
10. sin
()
Give orally the general measures
sin
()
=
-1.
18. cos
360°:
11.
7. cos = O.
2'
-1.
angles less than
()
6. cos = - ~.
tV2.
4. sin () =
16.
8 - 2 = 0,aquadraticequationintan28.
tan2 8
tan 8
= 0,
EXERCISES
.'. 8 = 90° and 36° 52.2'.
+
in terms of cot 8,
1
cot 8 + cot 8 = 2.
Solving for cot 8, cot 8 = 1.
. . . 8 = cot-1 1 = tll" or tll".
.
tan 8 VI
It is
tan 8 + cot 8 = 2; solve for 8 < 211".
Solution.-Expressing
2. cos
Example 4.-Given
73
()
14.
= -!V3.
of the following
=
- ~.
( -~ ~)-
()
16. = cos-1 (-!V2).
angles:
20. =
()
tan-1
f.
21. () = sin-l (-!V3).
19. () = tan-l V3.
= 1.
()
Solve the following for values of
< 360°:
22. tan () = -0.69321.
Ans. 145° 16' 11", 325° 16' 11".
17. cos
()
23. cos
24. cos
()
26.
()
()
= -0.27689.
Ans. 10604' 28", 253° 55' 32".
= :to.89613.
Ans. 260 20' 46", 153° 39' 14", 206° 20' 46", 333° 39' 14".
sin
26. cot ()
=
:to.80001.
Ans. 53° 7' 53", 126° 52' 7", 233° 7' 53", 306° 52' 7".
Ans. 24° 38' 26", 204° 38' 26".
1.2345.
Ans. 50° 59' 26", 230° 59' 26".
= 2.1801.
27. tan () =
28. cos () = :to.73218.
Ans. 42° 55' 51", 137° 4' 9", 222° 55' 51", 317° 4' 9".
PLANE AND SPHERICAL
74
29. sin e
=
TRIGONOMETRY
FUNCTIONS
:1:0.29868.
Ans. 17° 22' 42", 162°37' 18", 197°22' 42", 342° 37' 18".
36.
37.
38.
39.
40.
Ans. 50° 46' 21", 230° 46' 21".
cot e = 0.81638.
sin !/I = !.
Ans. 60°, 300°.
cos 2/1 = !0.
Ans. 22° 30', 157° 30', 202° 30', 337° 30'.
Ans. 15°, 75°, 135°, 195°, 255°, 315°.
tan 3/1 = 1.
Ans. 30°, 60°, 120°, 150°, 210°, 240°, 300°, 330°.
sec 2/1 = :1:2.
Ans. 20° 37' 14", 69° 22' 46".
sin 2/1 = 0.65923.
200° 37' 14", 249° 22' 46".
Ans. 109° 13', 250° 47'.
cos !/I = :1:0.57916.
Ans. 54° 46' 20".
tan !/I = 0.51804.
Ans. 135°, 315°.
sin /I = -cos /I.
Ans. 45°, 135°, 225°, 315°.
tan /I = cot /I.
Ans. 210°, 330°.
2 sin 2 /I - 3 sin /I = 2.
41.
4 cos2 /I
30.
31.
32.
33.
34.
35.
42. 2 sin2 /I
+ 20
cos /I
+ 3 sin
/I
+
2 cos /I
=
+ 0.
Ans. 60°, 135°, 225°, 300°.
Ans. 210°, 270°, 330°.
1 = O.
43. 2 cos2 /I + v'3 cos /I = 3(v'3 + 2 cos /I).
+ tan2 /I.
cos /I + sin /I = 2.
45. v'3
46.
2 cos2 /I
+
Ans. 150°, 210°.
Ans. 45°, 135°, 225°, 315°.
44. csc2 /I = 1
Ans. 30°.
11 cos e = 6.
Ans. 60°, 300°.
47. 2 cos2 /I + sin /I = 1.
Ans. 90°, 210°, 330°.
Ans. 30°, 45°, 135°, 150°, 225°, 315°.
48. cos 2/1(1 - 2 sin /I) = O.
49. cos /1(3- 4 sin2 2/1) = O.
Ans. 30°, 60°, 90°, 120°, 150°, 210°, 240°, 270°, 300°, 330°.
+ 1 = v'3 + cot e.
Ans. 45°, 150°, 225°, 330°.
51. Eliminate /I from the equations sin3 /I = x, and cos3 8 = y.
50. v'3 tan /I
lIns. xi
Suggestion.-Find
sin2 /I and cos2 /I and add.
52. Eliminate /I from the equations
acos/l+bsin/l=c
d cos /I + e sin /I =
f.
Ans. (bf - ce)2+
(cd
-
af)2
= (bd - ae)2.
Suggestion.-Solve for sin e and cos /I.
'
..
.
2 /I + cos2/I 1.
cd - af cos /I = -bf - ce ; su b stltute
l
ese
va
ues
IDSID
th
=
SID /I =
bd -ae
bd -ae '
y;
solve
for
rand
8.
53. Given r cos /I = x, and r sin /I =
Ans. r = VX2 + y2; /I = tan-1 J1.
x
x, r cos /I cos rp = y, r sin rp = z; solve for r,
.
-
54. Given r sin e cos rp =
/I, and rp.
Ans. r = VX2 + y2 + Z2;rp = sin-l
.
55. Fmd the value of:
sin
(
60°)
CDS150° +
Z
V + y2 +
cos (-
X2
60°)
sin 150° +
Z2
cot (-
; /I = tan-l~.
Y
60°)
tan 150°
.
Ans.3.
OF LARGE
ANGLES
75
56. Prove that tan 230° . cot 130° . sec 310°
tan 130° cot 230° csc 410° = tan 50°.
57. If
sec 230° sin 320°
~
=
cot 220°
a
(!..- + e) = H, show that cot (..- - /I)
= -f~
SID
130 ° - a, s h ow th a t
'
.
58. If cos
and csc (..- + 8)
= -H.
In the following problems, find all values of /Iless than 360°. Check each
angle.
59. sin 3/1
v'3
= - 2'
60. cos 3/1
= -~2"
61. cot 38 = -1.
62. sec 4/1 = 2.
l
GRAPHICAL REPRESENTATION
77
OM
OM
cos () = OM
OP = OH -- 1 = OM.
CHAPTER
GRAPHICAL
VI
REPRESENTATION
OF TRIGONOMETRIC
FUNCTIONS
56. Line representation
of the trigonometric
functions.Construct a circle of radius OH, with its center at the origin of
coordinates (Fig. 54). Since, in finding the trigonometric functions of an angle with its vertex at the origin of coordinates and
its initial side on the positive part of the axis of abscissas, any
Stated in words these are as follows:
The sine of an angle () is represented by the ordinate of the point
where the terminal side cuts the circumference of the unit circle.
The cosine of an angle () is represented by the abscissa of the point
where the terminal side cuts the unit circle.
It should be noted that the ordinate gives the value of the sine
both in magnitude and in sign. That is, when the point is above
the x-axis, the sine is positive, and when below it is negative;
likewise, for the cosine with reference to the y-axis. In this way
one can visualize the sine and the cosine.
Draw tangents to the circle at Hand E (Fig. 54), to meet the
terminal side OP extended or produced back through the origin
as the position of the angle requires.
In each of the four figures,
triangles OMP, OHD, and OEF are similar. Assume that HD
is positive when measured upward, and negative when measured
downward; also that EF is positive when measured to the right,
and negative when measured to the left.
From the similar triangles,
FIG. 54.
point may be chosen in the terminal side of the angle, we may
take the point where the terminal side cuts the circumference
of the circle.
Draw
angle
()
=
angle
XOP
in each of the four
quadrants, and draw MP ..L OX in each case. Now choose OH
as the unit of measure, that is, OH = 1. Then in each of the
four quadrants,
.
MP
MP
MP
SIn () =
= - 1 = MP .
OP = OH
~
76
MP
HD
OM
EF
and
OM = OH
MP = OE'
2~lP HD
HD
tan () =
OM = OH - 1 = HD.
OM
EF
EF
cot () =
MP = OE - 1 = EF.
Or, in words, these are:
The tangent of an angle () is represented by the ordinate of the
point where the terminal side of () is cut by a tangent line drawn to
the unit circle where the circle cuts the positive part of the axis of
abscissas.
The cotangent of an angle () is represented by the abscissa of the
point where the terminal st'de of () is cut by a tangent line drawn to
the unit circle where the circle cuts the positive part of the axis of
ordinates.
Let it be assumed that OD and OF are positive when measured
on the terminal side OP of the angle, and that they are negative
when measured on OP produced back through the origin. Then
in each of the four quadrants,
78
PLANE
AND
SPHERICAL
OP
OD
O~
OM = OH - 1 = OD.
OF
OP
OF
csc (J = MP = OE - 1 = OF.
sec
(J
=
Or, in words, these are:
The secant of an angle (Jis represented by the segment of the terminal side of (Jfrom the origin to the point where the line representing
the tangent of (Jcuts the terminal side.
The cosecant of an angle (J is represented by the segment of the
terminal side of (Jfrom the origin to the point where the line representing the cotangent of (Jcuts the terminal side.
It is not to be understood that the functions are lines; but that,
where the radius is taken as the unit of measure, and the lines are
expressed in terms of this unit, the numbers which then represent the lines are the functions.
Thus, if MP (Fig. 54) is 4 in.
and the radius is 7 in., MP in terms of OH is t, which is then
the sine of (J.
Historically, the line definitions of the trigonometric functions
were given before the ratio definitions.
This graphical way of
representing the functions assists in clarifying many questions
arising in connection with the functions.
For instance, it makes
apparent the origin of the terms tangent and secant of an angle.
This manner of defining; ~ function" ga ve
,y
rise to the term circular functions by which
they are often called.
EXERCISES
x~
y'
FIG. 55.
GRAPHICAL
TRIGONOMETRY
0 to -1.
From 270 to 360°, the ordinate increases from -1 to O.
Therefore as the angle varies from 0 to 360°, the sine varies from
0 at 0° to 1 at 90°, to 0 at 180°, to -1 at 270°, and back to 0 at
360°.
The cosine, being represented
by the abscissa of the point where
the terminal
side of the angle intersects
the unit circle, will
then decrease from 1 to 0 as the angle increases from 0 to 90°.
From 90 to 180°, the cosine is negative
and decreases
from
0 to - 1. From 180 to 360°, the cosine increases
from - 1
through 0 at 270° to 1 at 360°.
EXERCISES
Discuss orally the changes in the following functions as 0 varies from 0
to 360°:
1. sin 20.
7. cos 30.
13. cot 20.
2. sin 30.
8. cos !o.
14. sec o.
3. sin 40.
9. cos (-0).
16. sin (0 + 30°).
4. sin !O.
10. tan O.
16. cos (0 + 45°).
6. 2 sin O.
11. tan 20.
17. sin (0 - 45°).
6. cos O.
12. 2 tan O.
18. sin (0 + a).
19. Trace the changes in sin 2 a as a varies from 0 to 2....
20. Trace the changes
The minimum value?
in sin a
+ cos
a.
What
is the maximum
value?
Find the values of a for these values of sin a + cos a.
+ cos a = O?
TRIGONOMETRIC
CURVES
For what values of a is sin a
68. Graph of y = sm a.-The
changes whlch taKeplace~-sin (J, as indicated in the preceding article, are best shown by a
YI
A
Draw the following angles and represent their
trigonometric functions as lines:
1. 30°.
3. 245°.
6. 90°.
2. 160°.
4. 330°.
6. 180°.
67. Changes in the value of the sine and cosine as the angle
increases from 0 to 360°.-Draw
a circle with unit radius (Fig.
55) and construct an angle (Jin each of the four quadrants.
Since
in a unit circle the sine of an angle (Jis represented by the ordinate
of the point where the terminal side of the angle intersects the
circle, the variation in the ordinate will represent the variation
in the sin (J. At 0° the ordinate is O. As the angle increases
from 0 to 90°, the ordinate increases from 0 to 1. As (Jincreases
from 90 to 180°, the ordinate decreases from 1 to O. From
180 to 270°, the ordinate becomes negative and decreases from
79
REPRESENTATION
211'
y'
X
Y=8in8
FIG. 56.
graph. Referring again to Fig. 55 (Art. 67), let OA be the unit
of measure.
Then the complete circumference is the measure
of 360°, that is, 360° may be represented by a line 211'units in
length. Layoff OB = 6.2832 on OX (Fig. 56). OB is then the
radian measure of 211',or OB = 211'. Then layoff the proportional
parts as indicated in the figure, using multiples of 111' and t1l'
only. (Other angles could be used as well as, or in addition to
80
GRAPHICAL
PLANE AND SPHERICAL TRIGONOMETRY
these, making the curve more nearly accurate; but for our purpose the easy proportional parts of 211"are used.) Layoff OA on
the y-axis. This will represent the unit for plotting the sines of
the angles.
Select various values of (Jfrom 0 to 211",determine the corresponding values of y, and plot the points of which these values are
the coordinates.
Values of (J:
Values of y:
Points:
0 i..- i..- 1"- !..- ~..- i..O! !V2 h/31
!v3!V2!
Po
0 p, P2 P3 p, P.
g
t..-!..0 -! -!V2
P7 P. Pg P,o
etc.
etc.
etc.
Draw a curve through these points.
The curve is the graph of
y = sin (J. It shows the change in sin (Jas the angle changes from
0 to 211".
It is evident that the curve will repeat its form if (Jwere given
values from 211"to 411",from 411"to 611",etc., or from 0 to -211",
etc. The curve is then periodic.
Here the angle and the function are both plotted to the same
unit or scale, that is, the unit on the y-axis is the same length as
that to represent 1 radian on the x-axis. The curve so plotted
may be called the proper sine curve. Often, however, for
convenience when plotting on coordinate paper, the angles are
plotted according to the divisions on the paper. For example,
1 space = 6° or 10°, or some other convenient angle, depending
on the size 01 the plot.
69. Periodic functions and periodic curves.-In
nature, there
are many motions that are recurrent.
Sound waves, light
waves, and water waves are familiar examples.
Motions in
machines are repeated in a periodic manner.
The vibration of
a pendulum is a simple case, as is also the piston-rod motion in \
an engine. Other familiar illustrations are the vibration of a
piano string, breathing movements, heart beats, and the motion
of tides. An alternating electric current has periodic changes.
It increases to a maximum value in one direction, decreases to
zero, and on down to a minimum, that is, to a maximum value
in the opposite direction, rises again, and repeats these changes.
It is thus an alternating current passing from a maximum in one
direction to a maximum in the other direction, say, 60 times
a second.
Before physical quantities that change in a periodic fashion
can be dealt with mathematically, it is necessary to find a mathematical statement for such a periodic change.
REPRESENTATION
81
Definitions.-A
curve that repeats in form as illustrated by
the sine curve is called a periodic curve. The function that
gives rise to a periodic curve is called a periodic function.
The
least repeating part of a periodic curve is called a cycle of the
curve. The change in the value of the variable necessary for a
cycle is called the period of the function.
The greatest absolute
value of the ordinates of a periodic function is called the amplitude of the function.
Example I.-Find
the period of sin n(J, and plot y = sin 2(J.
Since, in finding the value of sin n(J, the angle (Jis multiplied
by n before finding the sine, the period is 211".
n
The curve for y = sin 2(J is shown in Fig. 57.
The period of
the function is 11"radians, and there are two cycles of the curve in
211"radians.
y
2
y
x
x
-1
-1
-2
FlU. 57.
F,c;, 58,
The number n in sin n(J is called the periodicity factor.
Example 2.-Find the amplitude of b sin (J,and plot y = 2 sin (J.
Since, in finding the value of b sin (J, sin (Jis found and then
multiplied by b, the amplitude of the function is b, for the greatest
value of sin (Jis 1.
The curve for y = 2 sin (Jis shown in Fig. 58. The amplitude
is 2.
The number b in b sin (J is sometimes called the amplitude
factor.
By a proper choice of a periodicity factor and an amplitude
factor, a function of any amplitude and any period desired can be
found.
While the sine function is perhaps the most frequently used of
the periodic functions, the cosine function can be used quite as
readily. By a proper choice and combination of sines and cosines
a function can be built up that will represent exactly or approxi-
GRAPHICAL REPRESENTATION
82
mately any periodic phenomenon.
Just how this may be done can
hardly be explained here.
60. Mechanical construction of graph of sin 6.-0n
one of
the heavy horizontal lines of a sheet of coordinate paper, choose
an origin and layoff angles every 15° from 0 to 360°, using each
small space to represent 15°, as in Fig. 59. With any convenient
point on this horizontal axis as a center, describe a circle with a
radius equal to 30 spaces. Choose the initial side of all the angles
on the axis of the angles.
j
eV
VI ;~I'\
eV
/,N
I
~~~I--
/
\
1\
/
15" 30"
45"
60"
75"
1)0"105" 120"
I
11 1
\--t
!
!
i
?\L\ 1\
d7
1
I
1
I
~i"
~rr-f!
Y I \ '~ ~: ~LV
~~!/
1/
f\.\1/1 I: I
i~
T..
t'-..... I!
I Ii'
~It
FIG.
Sine from curve
195" 210" 225" 240" 255" 270" 285" 300"
~~"60\",,\
~Y 'Y II \\ "
I~~
I
~I\Jy,
1-v
f' 1-1/1 \
Sine from table
Difference
I
EXERCISES
1. Plot
y = sin IJ, first, using as a unit on the x-axis a length twice as
great as that on the y-axis; second, using as a unit on the x-axis a length
one-half as great as that on the y-axis.
Plot both curves on the same set
of axes.
2. Plot y = cos e. Give its period and amplitude.
3. Plot y = tan IJand y = cot IJon the same set of axes.
4. Plot)/ = sec IJand y = csc IJon the same set of axes.
6. Plot Y = sin x + cos x.
Suggestion.-Plot
YI = sin x and Y2 = cos x on the same set of axes.
59.
Then
By means of the protractor layoff the central angles every 15°
from 0 to 360°, such as L.AOB, L.AOe, etc. Let the radius of
the circle be the unit of measure.
Then the sines of the angles
are the ordinates of the points A, B, e, etc. Through B draw a
horizontal line to intersect the vertical line through 15° as plotted
on the horizontal axis. The point b, thus determined, has as
coordinates (15°, sin 15°). In the same way locate c (30°, sin 30°);
d (45°, sin 45°); e (60°, sin 60°); etc. Through these points
draw a curve.
With the sine curve thus constructed, one can determine the
value of the sine of an angle approximately by measurement.
By measurement, the ordinate for
For example, find the sin 51°.
sin 51° is 23.3 spaces. Since the unit is 30 spaces,
0.7766.
18°
57°
78°
99°
123°
138°
171°
Dl ciB)~A
.~Fl-E(
=
2;03
I
~N~
\ I ,;1 //11
Hf
~"\\ 1/ ~~Vi ~L--
;- -
=
From the table of natural functions sin 51° = 0.77715. A
comparison of the results with Table IV for a number of angles
will give an idea of the accuracy of the graph.
Exercise.-Measure
the ordinates for the angles given in the
following table, compute the sines, and tabulate the results.
Find the sines of the same angles from the Tables and tabulate.
Compare the results.
~E
II 71 "'.p
I
\/ \'\
V F1\
""
sin 51°
IJ
J
1
al
0"
83
AND SPHERICAL TRIGONOMETRY
PLANE
y
find the points
= YI
+
6. Plot
on the curve
y
= sin
x
+
cos x from the relation
Y2, by adding
the ordinates
for various
values of x.
y
= sin2 x and y = cos' x on the same set of axes.
the curves never extend below the x-axis.
7. Plot Y =
t
Note that
sin x, y = sin x, y = 2 sin x, and y = ! sin x on the same
set of axes.
Give the period and the amplitude of each.
8. Plot y = sin tx, y = sin x, y = sin 2x, and y = sin ix on the same
set of axes.
Give the period and the amplitude of each.
61. Projection
of a point having
uniform
circular
motion.
Example I.-A point P (Fig. 60) moves around a vertical circle of
radius 3 in. in a counterclockwise direction.
It starts with the
point at A and moves with an angular velocity of I revolution in
10 sec. Plot a curve showing the distance the projection of P on
the vertical diameter is from 0 at any time t, and find its equation.
84
PLANE
AND
SPHERICAL
Plotting.-Let OP be any position of the radius drawn to the
moving point. OP starts from the position OA and at the end
of 1 sec. is in position OPI, having turned through an angle of
36° = 0.6283 radian. At the end of 2 sec. it has turned to OP2,
through an angle of 72°
OP3,OP4,
. . . ,OPIO.
=
GRAPHICAL REPRESENTATION
TRIGONOMETRY
1.2566 radians, and so on to positions
. . . ,
The points NI, N2, . . . are the projections of PI, P2,
respectively, on the vertical diameter.
Produce the horizontal diameter OA through A, and lay
off the seconds on this to some scale, taking the origin at A.
For each second plot a point whose ordinate is the corresponding distance of N froin O. These points determine a curve of
y
85
Similarly, the projection of the moving point upon the horizontal is given by the ordinates of the curve whose equation is
y
= r cos wt.
If the time is counted from some other instant than that from
which the above is counted, then the motion is represented by
y = r sin (wt + a),
where a is the angle that OP makes with the line OA at the
instant from which t is counted.
As an illustration of this
consider the following:
Example 2.-A crank OP (Fig. 61) of length 2 ft. starts from
a position making an angle a = 40° = ~7r radians with the
horizontal line OA when t = O. It rotates in the positive
direction at the rate of 2 revolutions per second. Plot the curve
y
~
FIG. 60.
Fw.
which any ordinate y is the distance from the center 0 of the
projection of P upon the vertical diameter at the time t represented by the abscissa of the point.
It is evident that for the second and each successive revolution,
the curve repeats, that is, it is a periodic curve.
Since the radius OP turns through 0.6283 radian per second,
and
or
angle AOP = 0.6283t radian,
ON = OP . sin 0.6283t,
y = 3 sin 0.6283t is the equation of the curve.
In general, then, it is readily seen that, if a straight line of
0, and
length r starts in a horizontal position when time t =
revolves in a vertical plane around one end at a uniform angular
velocity w per unit of time, the projection y of the moving end
upon a vertical straight line has a motion represented by the
equation
y = r sin wt.
f\1.
showing the projection of P upon a vertical diameter, and write
the equation.
Plotting.-The
axes are chosen as before, and points are found
for each 0.05 sec. The curve is as shown in Fig. 61.
The equation is y = 2 sin (47rt + 171").
Definitions.-The
number of cycles of a periodic curve in a unit
of time is called the frequency.
It is evident that
1
f =-,T
where f is the frequency and T is the period.
In y = r sin (wt + a), f = w and T = 271".
271"
w
The angle a is called the angle of lag.
62. Summary.-In
summary it may be noted again that
the equation
y = a sin (nx + a)
GRAPHICAL
86
PLANE
AND
TRIGONOMETRY
SPHERICAL
gives a periodic curve. In this equation there are three arbitrary constants, a, n, and a. A change in anyone of these
constants will change the curve.
(1) If a is changed, the amplitude of the curve is changed.
(2) If n is changed, the period of the curve is changed.
(3) If a is changed, the curve is moved without change in shape
from left to right or vice versa.
63. Simple harmonic motion.-If
a point moves at a uniform
rate around a circle and the point be projected on a straight
line in the plane of the circle, the oscillating motion, that is,
the back-and-forth
motion, of the projected point is called
simple harmonic motion. The name is abbreviated
s.h.m.
In Art. 61, the point N of Fig. 61 is the projection of the point
P. As P moves around the circle the point N moves back-andforth along the vertical diameter and performs a simple harmonic
motion.
It is readily seen that the point N moves more slowly
near the ends of the diameter and more rapidly near the center.
It thus changes its velocity or is accelerated.
It can be shown that many motions that one wishes to deal
with are simple harmonic.
Such is the motion of a swinging
weight suspended by a string, a pendulum, a vibrating tuning
fork, the particles of water in a wave, a coiled wire spring supporting a weight when the weight is pulled downward and
released.
Also many motions which are not simple harmonic
may be treated as resulting from several such motions.
Such
motions occur in alternating electric currents, in sound waves,
and in light waves.
EXERCISES
=
Ans.
3. Plot the curves that
represent
(a) y
(b) y
Give the period and frequency
=
=
the following
10 sin (4t
4 sin (vt
of each.
4. Plot y = r sin 1rt and y = r sin (1rt + tn-) on the same set of axes.
Notice that the highest points on each are separated by the constant angle
tw.
motions:
+ 0.6);
+ T\-1r).
(a) 1.571, 0.637; (b) 16,~.
Ans.
Such
curves
are
said
to be out
of phase.
The
stated in time or as an angle.
In the latter case it
6. Plot y = r sin i1rt, y = r sin (twt - tw), and
same set of axes.
What is the difference in phase
6. What is the difference in phase between the
y
= cos x? Between y = cos x and y = sin (x +
PRINCIPAL
VALUES
OF INVERSE
difference
in phase
is
is called the phase angle.
y = r cos i1rt all on the
between these?
curves of y
= sin x and
!1r).
FUNCTIONS
64. Inverse functions.-We
have seen that sin-1 t means the
angle whose sine is t. In Art. 57, it was shown that the sine
function varied from -1 to +1. Then the equation e = sin-1 t
has real solutions when and only when t is not less than -1
or greater than + 1. In the same way it can be shown that
e = cos-1 t has a solution when and only when t is not less than
-1, or greater than +1.
Since tan e and cot e can have any value from - 00 to + 00,
the equations e = tan-1 t and e = cot-1 t have solutions for all
values of t.
The two expressions sin e = t and e = sin-1 t both express
the same thing, namely, that e is an angle whose sine is equal to
t. In the first expression t is a function of e and in the second
e is a function of t.
In sin e = t, there is but one value of t for every value of e.
.
sin () is then said to be a single valued function of e.
In e = sin-1 t for every value of t, there are an indefinite
number of values of e, as was seen in Art. 53. sin-1 t is then said
to be a multiple valued function of t.
65. Graph of y = sin-1 x, or y = arc sin x.-Stating
y = sin-1 x
in the form sin y = x, it is readily seen by comparison with
y
1. A crank 12 in. long starts from a horizontal position and rotates in the
positive direction in a vertical plane at the rate of 21r radians per second.
The projection of the moving end of the crank upon a vertical line oscillates
with a simple harmonic motion.
Construct
a curveAns.thaty represents
this
= 12 sin 21rt.
motion,
and write its equation.
with
2. A crank 6 in. long starts from a position making an angle of 55°
the horizontal,
and rotates in a vertical plane in the positive direction at
the rate of 1 revolution in 5 sec. Construct a curve showing the projection
of the moving end of the crank on a vertical line. Write the equation of the
55); 5; i.
curve and give the period and the frequency. y
6 sin (t. 72° +
87
REPRESENTATION
=
sin x, that
sin y = x is obtained
from
y = sin x by inter-
changing x and y. Then the graph of y = sin-1 x is obtained
by plotting the sine curve on the y-axis instead of the x-axis as
in Art. 58. The curve is shown in Fig. 62.
In many mathematical operations where sin-1 x enters, it is
often desirable and, indeed, necessary to consider a portion of
the curve (Fig. 62, for which there will be but one value of y
for every value of x.
A glance at the figure will show that for the portion AOe of the
curve the function is single-valued.
That is, for every value of x
between and including -1 and + 1, y takes values between and
including -!1l" and h.
88
PLANE
AND
SPHERICAL
GRAPHICAL REPRESENTATION
TRIGONOMETRY
Definition.-The
values of sin-1 x between and including -tnand tn- for each value of x are called the principal values of sin-1 x.
To represent the principal value of the function, the s is often
written a capital, thus, Sin-1 x. The other functions are denoted
in a similar manner.
Yl~
The notation Sin-1 x denotes the principal
values of sin-1 x. The values are from - tnto !11".
The notation COS-l x denotes the principal
values of cos-1 x. The values are from 0 to 11"
c
inclusive.
The notation Tan-1 x denotes the principal
values of tan-1 x. The values are from - tnto !11".
X
The notation
Cot-l denotes the principal
1n--~A
values of cot-l x. The values are from 0
to 11".
Example.-Evaluate.
A
~i"
Ctxy152
9x2
+ 15
sin-1 tx) J:.
Solution.-The
notation is explained in the
solution to Example 4, page 69
Substituting x = 5,
~
Q
![sin-l 1 tan-l -1)
17. 16[sin-1(-0.2) .- sin-l (-0.4)
18. tan-I! + tan-1 ( -1).
20.
(-D + cos-l 2va .
x
Ans. O.
5".
Ans. 6'
cos-10 - tan-l -0)
21. cos-1
( - ~) + sin-l
22. cot-l ( - 0)
23. sin-1
+
( - ~3)
(-1)
(-
sin-1
Ans.
~2)
Ans. -3'4".
24. 8[cos-1(0.2) - cos-l (0.4)].
25. tan-l (- Ja)-tan-l
Ans.
(-0)
26. Plot y = cos-1 x and show that for values of y from 0 to .". inclusive
the values of x range from + 1 to -1, inclusive.
.
X 2
27. Evaluate sm-l
Ans. "..
2"J -2
28. Evaluate~(xv'9=X2
4"'b[~(X~
30. Evaluate
lO(x~
23.5620 - 11.6719 = 11.890.
Ans.
EXERCISES
Give the results of the following orally:
6. Sin-1 (-!).
1. COS-l ( -1).
7. Arc cot i0.
2. Sin-1 ( -1).
11. tan Cot-l V3.
12. cos Sin-1 (-~)
3. Sin-l~.
8. Arc tan V3.
13. sin COS-l ~.
4. Cos-110.
9. Arc cos (-10).
14.. sin COS-l (-!)
5. COS-l (-lVa).
10. Arc sin !V3.
15. cos Sin-1 1.
In the following find the numerical values of the given expressions, using
the principal values of the angles. In many applications of anti-functions,
as in the calculus, they enter into the expressions for areas, volumes, etc.,
and the angles must be expressed in radians.
1) vers-l
x
+ a2
+ a2
+
_
y2x
Ans. 9"..
4
nJ:
+ 9 sin-l
29. Evaluate
32.
! 2-\/225 - 9 .22 + 15 sin-1 i
= 5.4991 + 6.1728 = 11.6719.
1.681.
Ans. ij
sm-
/
2,
11'"
3'
Ans. 1.833.
- cos-1(-1.)
23.5620
Substituting
Ans. 9".
ZO'
Ans. 3.363.
Ans. -0.3217.
16.
19. sin-1
89
sin-1
~)ta
D J:.
Ans.
2".2a2b.
Ans. 5a2"..
1
-
x2
1
J0
Ans. 1.
PRACTICAL
CHAPTER
PRACTICAL
APPLICATIONS
VII
AND RELATED
PROBLEMS
66. Accuracy.-It
is of very great importance that one should
bear in mind as far as possible the limitations as regards accuracy.
The degree of accuracy that can be depended upon in a computation is limited by the accuracy of the tables of trigonometric
functions and logarithms used, and by the data involved in the
computation.
The greater the number of decimal places in the table, the
more accurately, in general, can the angles be determined from
the natural or logarithmic functions; but, in a given table, the
accuracy is greater the more rapidly the function is changing.
Since the cosine of the angle changes slowly when the angle is
near 0°, small angles should not be determined from the cosine.
For a like reason, the sines should not be used when the angle is
near 90°. The tangent and the cotangent change more rapidly
throughout the quadrant and so can be used for any angle.
Most of the data used in problems are obtained from measurements made with instruments
devised to determine those
-
APPLICATIONS
AND
RELATED
PROBLEMS
91
(1) Distances to two significant figures, angles to the nearest
0.5°.
(2) Distances to three significant figures, angles to the nearest
5'.
(3) Distances to four significant figures, angles to the nearest
1'.
(4) Distances to five significant figures, angles to the nearest
0.1'.
In drill problems, the angles are often expressed as if accurate
to seconds when the distances are expressed in five figures. This
gives variety in interpolating,
but one should not be misled
by the implied accuracy.
The United States Coast and Geodetic
Survey sets the following standards for its finest surveys: A line
1 mile long may turn to the one side or the other not more than
t in. The average closing error in leveling work must be less
than 1 in. in 100 miles. . The first gives a variation in the angle
of 0.4" to each side, or a total of 0.8". In making such accurate
computations, a 10-place table is used.
67. Tests of accuracy.-The
practical man endeavors in one
way or another to check both his measurements and his computations. In our work here we are interested in checks on the
computation.
(1) Often a graphical construction to scale will give results
s
data depends not only upon the instruments used, but upon the
person making the measurements and upon the thing measured.
A man in practical work uses instruments which are of such
accuracy as to secure results suitable for his purpose.
The data
given in problems for practice are supposed to be of such accuracy
as the instruments that are used in such measurements would
warrant.
In the solution of a problem it is useless to carry out the
computations with a greater degree of accuracy than that of the
data. That is, if the data are accurate only to, say, four significant figures, there is no necessity to compute accurately to more
figures than this. If the measuring instrument can be read only
to minutes of angle, in the computation there is no object in
carrying the work to seconds of angle.
In general, the following is the agreement between the measurement of distances and the related angles:
00
made free-hand, only the gross mistakes in computation will be
discovered; but if the construction is made carefully with accurate instruments, results may be obtained as accurate as the data
will warrant.
This, then, may be considered a graphical solution
of the problem.
(2) Mistakes in the computations may be found by making
another computation using a different set of data; or by recomputing, using the same data but using a different set of formulas.
Many ways will present themselves to the thoughtful student.
EXERCISES
1. In determining an angle by means of a table of natural functions that is
correct to five places, if the angle is near 1 can seconds be determined from
°
the cosine of the angle?
Can tenths of minutes?
Can minutes?
2. Answer the same questions as in Exercise 1 if the sine is used instead of
the cosine.
If the tangent is used.
If the cotangent.
S. Answer similar questions if the angle is near 89°, 80°, 10°,20°, 70°, 4.)°.
PRACTICAL
PLANE AND SPHERICAL TRIGONOMETRY
92
4. From the results obtained in the first three exercises, state
as to what sized angles can be determined most accurately from
tangent, and cotangent of the angle.
6. Compare the logarithms of 92.8766 and 92.876; 99.8375,
99.838; 121.575, 121.57, and 121.6.
6. Can a number be determined correct to six figures by using
logarithm table?
When?
When is it not possible to determine
of a number by means of a five-place table of logarithms?
APPLICATION
OF RIGHT
TRIANGLES
conclusions
sine, cosine,
99.837, and
a five-place
five figures
TO VECTORS
68. Orthogonal projection.-If
from a point P (Fig. 63a),
a perpendicular PQ be drawn to any straight line RS, then the
P
\\
R /Q
\/
s
.
0
(a)
B
/"""'i
_f:~--tx
I
C (b)
B
O~
~
A
~ll..x
--- (c)
APPLICATIONS
AND
RELATED
PROBLEMS
93
equals the length of the segment multiplied by the sine of the angle
of inclination.
69. Vectors.-In
physics and engineering, line segments are
often used to represent quantities that have direction as well as
magnitude.
Velocities, accelerations,
and forces are such
quantities.
For instance, a force of 100 lb. acting in a northeasterly
direction may be represented by a line, say, 10 in. long, drawn
in a northeasterly direction.
The line is drawn so as to represent
the force to some scale; here it is 10 lb. to the inch. An arrow
head is put on one end of the line to show its direction.
In Fig. 65, OP = v is a line representing a directed quantity.
Such a line is called a vector. 0 is the beginning of the vector
and P is the terminal.
OQ = x is the projection of the vector
I
JE
Y
FIG. 63.
foot of the perpendicular Q is said to be the orthogonal projection,
or simply the projection, of P upon RS.
The projection of a line segment upon a given straight line
is the portion of the given line lying between the projections of
the ends of the segment.
In Fig. 63b and c, CD is the projection of AB on Ox.
R
y
,P
0
lzL ~
~-/
R
v
,(j
I
I
x
Q
FIG. 65.
p
on the horizontal
'-'1)
X
OX, OR
o'
=
/
/
>Q
'
FIG. 66.
y is the projection
on the vertical
OY, and 0 is the inclination of the vector.
The vectors x and y
aw cal1(;d comoonents of the vector 1'. As befor
y
x = v cos 6,
cos O.
~
mp:B
[ IA
R~'
01
r
M
:
x
FIG. 64.
[9]
I
I
N
The projections usually made are
upon a horizontal line OX and a vertical
line OY, as in Fig. 64. Hence, if l is
the length of the segment of line proX jected, x the projection on OX, y the
projection on OY, and 0 the angle of
inclination, that is, the angle that the
line segment makes with the x-axis, then
x = 1 cos 6,
and
[10]
y
= 1 sin 6.
This may be stated in the following:
THEoREM.-The
projection of any line segment upon a horizontal line equals the length of the segment multiplied by the cosine
of the angle of inclination; the projection upon a vertical line
-
------
and
y = v sin 6.
Suppose the vectors OQ and OP (Fig. 66), represent the magnitude and direction of two forces acting at the point 0, and having
any angle cpbetween their lines of action.
If the parallelogram
OQRP is completed, then the diagonal OR represents in magnitude and direction a force that will produce the same effect as
the two given forces.
The vector OR is called the resultant of the vectors OQ and
OP. The process of finding the resultant of two or more given
forces is called composition of forces.
Conversely, the vectors OQ and OP are components of OR.
Since QR is equal and parallel to OP, it follows that the two
components and their resultant form a closed triangle OQR.
The relations between forces and their resultant may then be
--
94
PRACTICAL
PLANE AND SPHERICAL TRIGONOMETRY
found by solving a triangle which is, in general, an oblique
triangle.
Example I.-Suppose
that a weight W is resting on a rough
horizontal table as shown in Fig. 67. Suppose that a force of
10 lb. is acting on the weight in the direction OP, making an angle
of 20° with the horizontal; then the horizontal pull on the weight
is OQ = 40 cos 20° = 37 .588 lb., and the vertical lift on the weight
is OR = 40 sin 20° = 13.68 lb.
Example 2.-A car is moving up an incline, making an angle
of 35° with the horizontal, at the rate of 26 ft. per second. What
is its horizontal velocity?
Y'
Horizontal velocity = 26 cos
35° = 21.3 ft. per second.
Vertical
velocity = 26 sin
.20'
14.9 ft. per second.
w
.
Q X 35° =
0
~
FIG. 67.
1. Find
EXERCISES
the projection
of a line
segment 31.2 ft. long upon a straight line making an angle of 34° 16.4' with
Ans. 25.78.
the segment.
2. The line segment AB, 32.67 in. long makes an angle of 45° 23' with
the line OX.
Find the projection
on OX.
Find the projection
on OY
perpendicular
to OX and in the same plane as OX and AB.
Ans. 22.95; 23.26.
,teamer is moving S. 21° W. at the rate of 28 miles per hour.
How
( in a, smnneny dileniull 7
Ans. 10.03 miles per hour; 26.14 miles per hour.
is it moving in B wpstNlr dircctlon
4. The direction a force of 1800 lb. is acting, makes an angle of 26° 35'
with the horizontal.
Find the horizontal and vertical components
of the
Ans. 1610 lb.; 805.5 lb.
force.
5. A ship is sailing at 20.5 miles per hour in a direction N. 24° 35' E.
Find the northerly and easterly components of its speed.
Ans. 18.64 miles per hour; 8.528 miles per hour.
6. A force or 300 lb. is acting on a body lying on a horizontal plane, in a
direction which makes an angle of 20° with the horizontal.
What is the
force tending to lift the body from the plane?
Ans. 102.6 lb.
7. A body weighing 58 lb. rests on a horizontal table and is acted upon
by a force of 55 lb., acting at an angle of 27° 45' with the surface of the
table.
What is the pressure on the table?
Ans. 32.39 lb.
8. A body weighing 71 lb. rests on a horizontal table and is acted upon by
a force of 125 lb. acting at an angle of (-31° 30') with the surface of the
table.
What is the pressure on the table?
Ans. 136.3 lb.
9. The horizontal and vertical components
of a force arc respectively
234.5 and 654.3 lb. What is the magnitude
of the force, and what angle
does its line of action make with the horizontal?
Ans. 695.1 lb.; 70° 16.95'.
APPLICATIONS
AND RELATED PROBLEMS
95
10. The horizontal and vertical components
of a force are respectively
145.7 and -175.3 lb. What is the magnitude of the force, and what angle
does its line of action make with the horizontal?
Ans. 227.95 lb.; -50° 16.1'.
11. A river runs directly south at 5 miles per hour.
A man starts at the
west bank and rows directly across at the rate of 4.5 miles per hour.
In
what direction does his boat move?
Ans. 41° 59.2' with bank.
12. A ferryboat at a point on one bank of a river!
mile wide wishes to
reach a point directly across the river.
If the river flows 3.75 miles per hour
and the ferryboat can stearn 8.1 miles per hour, in what direction should the
boat be pointed?
Ans. 27° 34.7' upstream.
13. Two men are lifting a stone by means of ropes in the same vertical
plane.
One man pulls 143 lb. in a direction 40° from the vertical and the
other 130 lb. in a direction 45° from the other side of the vertical.
Determine the weight of the stone.
Ans. 201.47 lb.
14. Two forces of 245 and 195 lb. act in the same vertical plane upon a
heavy body, the first at an angle of 42 withO
the horizontal
and the second at an angle
of 60°. Find the total force tending to move
the body horizontally;
to lift it vertically.
Ans. 279.6 or 84.77 lb.; 332.8 lb.
USEFUL
AND MORE
PROBLEMS
DIFFICULT
70. Distance and dip of the horizon.
In Fig. 68, let 0 be the center of the
earth, r the radius of the earth, and h
Tl'lP l1el!!:ht ot
P
face; to find the distance from the
point P to the horizon at A.
Bygeometry,PA2
= P02 - OA2 = (r
.'. PA = v2rh
FIG. 68.
+ h)2
+ h2.
- r2
= 2rh
+ h2.
For points above the surface that are reached by man, h2 is
very small compared with 2rh,
... PA = ~
approximately.
In the above, P A, r, and h are in the same units. A very
simple formula can be derived, however, if h be taken in feet, r
and PAin miles, and r = 3960 miles. Then
PA = ~2 X 3960 X
52~0 = ~~h miles.
The following approximate rules may then be stated:
The distance of the horizon in miles is approximately equal to the
square root of ~ times the height of the point of observation in feet.
96
PLANE AND SPHERICAL
TRIGONOMETRY
The height of the point of observation in feet is i times the square
of the distance of the horizon in miles.
Definition.-The
angle APC = (J in Fig. 68 is called the dip
of the horizon.
Evidently,
tan
P
(J = A.
r
71. Areas of sector and segment.-Formulas
for solving for
the areas of the sector and segment of a circle are derived here
so that they may be used for reference.
From geometry, the area of the sector of a circle as XOA
(Fig. 69) equals the arc XnA times one-half the radius OX.
By Art. 8, arc XnA = OX X (J,where (Jis expressed in radians.
Hence, using r for the radius and S
for the area of sector,
S
=
!r2(J.
Evidently, the area of the segment
XAn
= S - area of triangle XOA.
X
But area of triangle XOA = !OXBA = !OX. OA sin (J = !r2 sin (J.
Hence, using G for area of segment,
G = !r2(J - !r2 sin (J.
[11]
.'. G = !r2(6 - sin 6).
0
FIG. 69.
As an exerci:,;e, the student may latcr show that this formula
holds when (Jis an obtuse angle. Also when 71"< (J < 271".
This is the simplest accurate formula for finding the area of
a segment of a circle. It is of frequent use in many practical
problems.
Various approximate formulas for finding the area
of a segment are given for the use of practical men not having a
knowledge of trigonometry.
Two of the best known of these
are the following:
= ihw
+
(1)
A
(2)
A = th2~2~
h3
Table
V, 78° 30'
APPLICATIONS
AND
Substituting
1.3701 radians.
PROBLEMS
97
0.9799.
these values in [11],
G
= ! X 162(1.3701 - 0.9799) = 49.94
.'. area of segment = 49.94 sq. in.
72. Widening of pavements on curves.-The
tendency of
a motorist to "cut the corners" is due to his unconscious desire
to give the path of his car around a turn the longest possible
radius. Many highway engineers recognize t.his tendency by
widening the pavement on the inside of the curve, as shown in
Fig. 70. The practice adds much to the attractive appearance
of the highway.
If the pavement is the same width around the
curve as on the tangents, the curved section appears narrower
than the normal width; whereas,
if the curved section is widened
gradually to the midpoint G of
the turn, the pavement appears
to have a uniform width all the
way around.
In order that the part added
may fit the curve properly, it is
necessary to have the curve of
the inner edge a true arc of a
FIG. 70.
circle, tangcnt to the edge of the
straightaway sections, and therefore it must start before the point
E of the curve is reached.
The part added may be easily staked
out on the ground with transit and tape, by means of data derived
from the radius r, the central angle (Jof the curve, and the width w.
In practice, the width w is taken from 2 to 8 ft. according to the
value of r. The area added can be readily computed when values
for r, w, and (Jare given.
Referring to the figure, derive the following formulas:
x
- 0.608,
=
RELATED
=
sin 78° 30'
2w '
where r is the radius, h the height of the segment, and w the
length of the chord.
Example I.-Fine
the area of the segment of a circle of radius
16 in., and having a central angle of 78° 30'.
Solution.-By
PRACTICAL
=
r sec !(J
-
r = --r-r
cos
x
+ w = r , sec
.'. r' =
-
1
2"(J
x 1+ w
see 2"(J- 1
t = r tan !(J.
t' = r' tan !(J.
r, =
=
-
(J
- r.
2"
r'
cos !(J
- r.,
ex + w) co: !(J
1
-
cos 2"(J
98
PLANE AND SPHERICAL
Area added
= BFCEAG = BPAO'
TRIGONOMETRY
-
FPEC - BGAO'.
BPAO' = r't'.
8
FPEC = FPEO - FCEO = rt - 3607172.
8
BGAO ' - 3607rr'2.
. . . area a dded
PRACTICAL
(Fig. 71), the length
taken in radians:
_I
L = 2v D2
-
r It I
-
(
= r't' - rt - 3:07r(rl
)
8
- 3607rr'2
+ r)(r' - r).
-
AND
RELATED
is given by the following
- r)2 +
(R
+ r)
7I"(R
PROBLEMS
formula
+ 2(R
-
99
where the angle is
R-r
r) sin-l-y;-.
13. Using the same notation as in Exercise 12, show that, when the belt is
crossed (Fig. 72), the length is given by the following formula:
L
8
r t - 3607rr2
APPLICATIONS
= 2VD2
-
(R
+
r)2
+
(R
+
r)(7I"
+
2 sin-1
R
i; r).
Note.-These formulas would seldom be used in practice. An approximate formula would be more convenient, or the length would be measured
with a tape line.
p
EXERCISES
1. A cliff 2500 ft. high is on the seashore.
How far away is the horizon?
Ans. 61.24 miles.
2. Find the greatest distance at which the lamp of a lighthouse can be
seen from the deck of a ship.
The lamp is 75 ft. above the surface of the
water and the deck of the ship 40 ft.
Ans. 18.4 miles.
3. Find the radius of one's horizon if located 1000 ft. above the earth.
How large when located 2.5 miles above the earth?
Ans. 38.73 miles; 140.7 miles.
4. How high above the earth must one be to see a point on the surface
35 miles away?
Ans. 816.7 ft.
6. Two lighthouses, one 100 ft. high and the other 75 ft. are just barely
visible from each other over the water.
Find how far apart they are.
Ans. 22.86 miles.
Ans. 1395 sq. ft.
7. A thin rope is fastened by its ends to two points 25 ft. apart and in a
horizontal plane.
It has a heavy weight hanging at its midpoint causing
it to sag 5 ft., and making the rope from center to ends extend in practically
straight lines.
Find the angle between one-half of the rope and a horizontal, and find the total length of the rope between the points of support.
Ans. 21 ° 48.1'; 26.93 ft.
8. The radius of a circle is 72.52 ft. In this circle a chord sub tends an
angle of 40° 32.4' at the center.
Find the difference between the length of
the chord and the length of its arc.
Ans. 1.066 ft.
9. Compute the volume for each foot in the depth of a horizontal cylindrical.oil tank of length 40 ft. and diameter 4 ft.
Ans. 98.27 cu. ft.; 251.33 cu. ft.; 404.39 cu. ft.; 502.66 cu. ft.
10. A cylindrical
tank in a horizontal
position is filled with water to
within 10 in. of the top.
Find the volume of the water if the tank is 10 ft.
long and 5 ft. in diameter.
Ans. 174.84 cu. ft.
11. Find the angle between the diagonal of a cube and one of the diagonals
of a face of the cube.
A ns. 35° 15.8'.
12. If Rand r are the radii of two pulleys, D the distance between the
centers, and L the length of the belt, shpw that, when the belt is not crossed
s
FIG. 71.
FIG. 72.
A rule often given for finding the length of an uncrossed belt is: Add twice
the distance between the centers of the shafts to half the sum of the circumferences of the two pulleys.
14. Using the formula of Exercise 12, and given R = 16 in., r = 8 in.,
Find the length by the approxiand D = 12 ft., find the length of the belt.
mate rule.
Ans. 30.32 ft.; 30.28 ft.
16. Use the same values as in Exercise 14, and find by the formula of
Ewrrisr
13 the length of the belt when crossed.
An". 30.52 ft.
16. An open belL connects
two pulleys of diameters
6 and 2 ft.,
respectively.
If the distance between their centers is 12 ft., find the length
of the belt.
ns. 36.9 ft.
D
B
"°A
FIG. 73.
17. Two pulleys of diameters 8 and 3 ft., respectively, are connected by a
crossed belt.
If the centers of pulleys are 14 ft. apart, find the length of the
belt.
Ans. 47.47 ft.
18. The slope of the roof in Fig. 73 is 33° 40'. Find the angle () which is
the inclination to the horizontal of the line AB, drawn in the roof and making
an angle of 35° 20' with the line of greatest slope.
100
PLANE
.
.
S 0 l utwn.-Slll
AND
SPHERICAL
TRIGONOMETRY
CB
AB'
AD
AD
.
or AB =
cos 35° 20' =
cos 35° 20'
AB'
0
=
or ED = CB = AD sin 33° 40'.
AD
sin 8 = AD sin 33° 40' +
cos 35° 20'
= sin 33° 40' X cos 35° 20'
= 0.55436 X 0.81580 = 0.45225.
sin 33° 40' = ~~,
Then
... 0 =
sin-l
0.45225
=
PRACTICAL
""
23. Show that placing the carpenter's
square as shown in Fig. 75b will
determine the miter for making a regular pentagonal frame as shown in a.
Ans. 0 = 54°.
What is the angle 0 of the miter?
C
AND
RELATED
PROBLEMS
101
bolted at points a, b, c, etc. These distances should be accurate to tenths of
an inch. Can these distances be determined by means of geometry?
Ans. Aa = 10 ft. 5.3 in., etc.; yes.
26. A -street-railway track is to turn a
r--"~--'1
/IN
M
corner on the arc of a circle. If the track is t. J
'"
at a distance a from the curbstone and the;
T
/f
turn is through an angle 8, show that the
radius r = OR = ON (Fig. 77) of the curve
to pass at a distance b from the corner is
given by the formula
i
26° 53.3'.
19. A hill slopes at an angle of 32° with the horizontal.
A path leads up
it, making an angle of 47° 30' with the line of
p
steepest slope; find the inclination
of the path
with the horizontal.
Ans. 20° 58' 40".
20. Two roofs have their ridges at right
51t
angles, and each is inclined to the horizontal
at an angle of 30°. Find the inclination
of
their line of intersection
to the horizontal.
4"
Ans. 22° 12' 28".
IB
5"
I
21.
A
mountain
side
has
a slope of 30°. A
I
"" I road ascending the mountain is to be built and
I is to have a grade of 7 per cent. Find the
iC angle it will make with the line of greatest
0
Ans. 81 ° 57.2'.
I slope.
I
22. Two set squares whose sides are 3, 4, and
I
5 in. are placed as in Fig. 74, so that their 4-in.
I
~i(le~ awl right angles coincide, aml tlw ~mglf'
A
between the 3-in. sides is 46° 35'. Find the angle
F IG. 74 .
0 between the longest sides.
Ans. 27° 26.9'.
APPLICATIONS
a - b cos !8
.
r =
1 - cos !8
27. When the 8-in. crank of a horizontal
engine is vertical, the piston is 1.5 in. past
the midstroke.
What is the length of the
connecting rod and what angle does the connecting rod make with the guides at this
instant?
Ans. 22.08 in.; 21 ° 14.6'.
28. In Fig. 78, LGA is an arc of a circle
with center at 0, LV and A V are tangents at
the extremities
of the arc, GF is tangent- to
the arc at its center point G, and 8 is the
angle at the center of the circle and intercepting
ing formulas useful in railway surveying:
t = r tan !8.
e = r (sec !8
"
-
1).
= 21 co::> !e.
c = 2r sin !8.
e = t tan l8.
GA. = ~csec 10.
29. A salesman for a wire-screen
a screen in the form of the frustum
Derive the follow-
the arc.
m = r vers !O.
c = 2m cot io.
company wishes formulas for laying out
of a right circular cone of large diameter
N
M
tj
~
J
81
A
-
(b)
B
A
FIG. 75.
0
FIG. 78.
0
FIG. 77.
24. If 12 in. is taken on one leg of a carpenter's
square, how many inches
must be taken on the other leg to cut miters for making regular polygons of
the following numbers of sides: 3, 4, 6, 8, and 10? Express results to the
Am. 20H; 12; 6H; 5; 3i.
sixteenth of an inch.
26. In the frame of a tower shown in Fig. 76, determine the distances from
A and B, C and D, etc., to make the holes in the braces so that they may be
D, small diameter d, and slant height s. He also wishes the dimensions
land w of the rectangular piece from which the screen is to be cut. The
layout is in the form of a section of a ring bounded by two concentric circles
of radii Rand
r, and having
a central
angle 8.
Determine
formulas
for R,
l
]02
PRACTICAL
PLANE AND SPHERICAL TRIGONOMETRY
r, and 0 in terms of D, d, and s; and formulas for land w in terms of R, r,
and o.
sD
sd
D - d
0 =-180°'
Ans . R = -'r
=-'
s
D - d'
D - d'
'
l = 2R sin !O;w = R - r cos !O.
REFLECTION AND REFRACTION OF LIGHT
73. Reflection of a ray of light.-The
path of a ray of light in
a homogeneous medium as air is a straight line. But when a ray
p
of light strikes a polished surface it is
R
S
reflected
according to the well-known
law
~
I
I
which states that the angle of incidence is
.
l
!
equal to the angle of reflection.
I
Thus, in Fig. 79, the incident ray SQ
strikes the polished surface at Q and is
FIG. 79.
reflected in the direction QR. The line QP
is perpendicular to the surface at Q. The angle SQP = i is the
angle of incidence, and the angle PQR = r is the angle of reflection. The law states that these two angles are equal.
74. Refraction of a ray of light.-When
a ray of light passes
from one transparent medium to another which is more or less
dense, its direction is changed, that is, the ray of light is refracted.
Thus, in Fig. 80, a ray of light SQ, passing through air, meets
p
the surface of a piece of glass at Q and is
S
I
toward
the
normal,
or
pcrpcndicul'e fracted
lar,
QP'.
It continues
in the
direction
QT
~
Air
~
until it meets the other surface of the glass
IQ
at T, where it is again refracted, but this
r
time away from the normal; and passes into
p;
the air in the direction TR. If the two
surfaces of the glass are parallel, it has been
found by experiment that the direction of
TR is the same as that of SQ.
FIG. 80.
The lines QP and QP' are perpendicular
to the surface at Q. The angle SQP = i is the angle of incidence,
and the angle P'QT = r is the angle of refraction.
It has been found by experiment that for a given kind of glass
the ratio
sin i
sinr=,u
.
.
is constant whatever the angle of incidence may be. This means
that, for a certain kind of glass, if the angle of incidence is
changed, then the angle of refraction also changes in such a
APPLICATIONS
AND
RELATED
PROBLEMS
103
manner that the ratio of the sines is constant.
This ratio for
a ray of light passing from air to crown glass is very nearly t, and
for water it is t.
The value of the ratio s~n i = ,uis called the index of refraction
sm r
of the glass with respect to air.
It follows that the index of refraction of air with respect to
glass is the reciprocal of that of glass with respect to air. That
is, if the index of refraction of glass with respect to air is ,u,
then the index of refraction
same may be stated
of air with respect
for any other
to glass is
two transparent
I.
The
,u
substances.
EXERCISES
1. Prove that if a mirror that is reflecting a ray of light is turned through
an angle a, the reflected ray is turned through an angle 2a.
2. The eye is 20 in. in front of a mirror and an object appears to be 25 in.
back of the mirror, while the line of sight makes an angle of 32° 30' with the
mirror.
Find the distance and direction of the
object from the eye.
A
Ans. 70.8 in. in a direction making an angle of
4° 3' with plane of mirror.
N
3. A ray of light passes from air into carbon
disulphide.
Find the angle of refraction if the
angle of incidence is 33° 10' and the index of
rf'fr:1ction is 1.758.
AIt~. 18° 7.9'.
4. When!, = 1.167 and the angle of incidence
M
is 19° 30', find the angle of refraction.
Ans. 16° 37.3'.
6. A ray of light travels the path ABCD
FIG. 81.
(Fig. 81) in passing through the plate glass MN
0.625 in. thick.
What is the displacement
CE if the ray strikes the glass at
an angle ABP = 42° 10' the index of refraction being ~?
Ans. 0.1887 in.
6. If the eye is at a point under water, what is the greatest angle from the
zenith that a star can appear to be?
Ans. 48° 35.4'.
7. A source of light is under water.
What is the greatest angle a ray can
make with the normal and pass into the air?
For any greater angle the
ray is totally reflected.
Ans. 48° 35.4'.
8. A straight rod is partially immersed in water.
The image in the water
appears inclined at an angle of 45° with the surface.
Find the inclination
of the rod to the surface of the water if the index of refraction is t.
Ans. 70° 31.6'.
SIDES OPPOSITE VERY SMALL ANGLES
75. Relation between sin 6, 6, and tan 6, for small angles.Draw angle BOE = () (Fig. 82). With 0 as a center and OB = 1
as radius, describe the arc BD. Draw DA 1. to OB and BE
PRACTICAL APPLICATIONS AND RELATED PROBLEMS
PLANE AND SPHERICAL TRIGONOMETRY
104
These results show that, for small angles, sin f)and tan f)may be
replaced by f) in radians and the results will be approximately
correct.
For example,
sin 5° 9.4' = 0.0899,
AD, f) = arc DB, and
tangent to the arc at B. Then sin f) =
tan f) = BE. Comparing areas of triangles and sector;
< sector OBD < bOBE.
bOBD
tOB2 . f), where
But bOBD = tOB X AD, sector OBD =. BE.
in radians (see Art. 8) and bOBE = tOB
Then
tOB' AD < tOB2. f) < tOB'
Dividing
by ! and substituting
1=
OB =
limit, written
~
Then, since sm
BE.
1, AD = sin f), and
= -cos f)'
that tan
<
f)
~
< sec
sm f)
N ow as f) approaches
limit sec f) approaches
f)
:~o sec
f)
and
sin f) < f) < tan f).
Dividing by sin f) and rem emsin f)
.
f)
1
FIG. 82.
is
BE = tan f),
benng
0
f)
f)
0 as a
1 as a
= 1.
is always
less than
a quantity
105
which
approaches 1 as a limit, and at the same time is greater than 1, we
5° 9.4' = 0.0900 radian,
tan 5° 9.4' = 0.0902.
For a smaller angle the agreement will be still closer.
76. Side opposite small angle given.- When a very small angle
and the side opposite are given in a right triangle, another side
can be found by means of the sine or tangent of the small angle
considered as a number of radians.
The short side, considered as
an arc, divided by the number of radians in the small angle will
give a long side.
Example I.-A tower is 125 ft. high. The angle of elevation
of the top of the tower, from a point in the same horizontal plane
as the base, is 1°. Find the distance from the point of observation to the tower.
Solution.-Let
x = distance to the tower in feet.
Then
But
125 or x = -. 125
tan 1° = -,
x
tan 1°
1° = 0.01745 radian = tan 1°, approximately.
r:;:
ij
lim
.'.
-.-------
8->0sin 6 = 1.
[12]
Again, dividing sin f) < f) < tan f) by tan f) and simplifying,
f)
lim
f)
1.
lim
f)
f
;
t
ere
ore,
Ih
cOS
8-+0tan f) =
=
<. 1 B ut 8-+0
cos f) <
tan f)
By computing the following table, the student will find [12]
verified
Angle in
degrees
\
\
20°
10°
5°
40
3°
2°
1°
~o
tan
0 in radians
sin 0
\
0
sin 0
0
\
x
=-
Example 2.-A
railway track has a 2 per cent grade for
a certain distance.
Find the inclination of the track to the
horizontal.
Solution.-The
per cent of a grade is the ratio of the number of
feet rise to the number of feet on the horizontal.
Then, for a 2
per cent grade the tangent of the angle of inclination is 0.02,
which is approximately the angle in radians.
By Table V, 0.02 radian = 1° 8.7'.
77. Lengths of long sides given.-In
a right triangle having
two sides, including an acute angle, given, the angle can be found
by the formula
.
tantA =
I
!
-
where A is the angle included
side b.
c=b
~-,c + b
by the hypotenuse
c and the
PRACTICAL APPLICATIONS AND RELATED PROBLEMS
PLANE AND SPHERICAL TRIGONOMETRY
106
This formula is derived as follows:
In Fig. 83, ABC is a right triangle.
Draw AE bisecting angle A, and draw DB perpendicular
Then!A
= LDAE = LCBD.
Also AD = AB = c, and CD = c - b.
.
In tnangle CBD, tan CBD
CD c - b
= Cl3 = ---a'
tan!A
+
= y(c
a = ~2
But
=
to AE.
b)(c - b).
[C=b
.'.tan!A
= '\jm'
When angle A is small, 2 tan!A gives approximately the value
of A in radians, which can be used as
B
already explained.
Example.-At
what distance may a
al \ E
mountain 1 mile high be seen at sea, if the
earth's radius is 3960 miles?
beD
Solution.-Let
8 = distance
in miles,
FIG.83.
and let the angle at the center of the
earth between the radius to the mountain and the radius to the
point at sea be e.
13961
- 3960
fl
= '\j792i'
By the formula, tan tje = ..J3961
+ 3960
Then
f} in radians
= 2~ 7912t'
By the formula 8 = rf} of Art. 8,
8 = 2~ 7~21 X 3960 = 89 miles.
This example can also be computed by the rule given in Art. 70,
which gives 8 = Vih, where h is the height of the mountain in
feet.
.'. yt X 5280 = 89 miles.
EXERCISES
10'. Find
1. A certain plane is inclined to the horizontal at an angle of 1 °
the per cent of grade of a railway track constructed
on this plane.
Ans. 2.04 per cent.
2. A railway track rises 100 ft. to the mile.
of the track.
Find the angle Ans.
of inclination'
1° 5.1'.
107
3. H the diameter of the earth as seen from the moon makes an angle of
1° 54', find the distance from the moon to the earth, taking the earth's
radius as 3960 miles.
Ans. 239,000 miles.
4. If the distance from the earth to the sun is 93,000,000 miles and the
diameter of the sun makes an angle of 32' at the earth, find the diameter of
the sun in miles.
Ans. 870,000 miles.
6. Telescopes at the ends of a base line 350 ft. long, on the deck of a ship,
are turned upon a distant fort.
The lines of sight of the telescopes are
found to make angles of 89° 10' and 89° 40' with the base line.
Find the
distance from the ship to the fort.
Ans. 3.26 miles.
6. The diameter of the moon sul:.tends an angle of 31' 5" at the earth.
The moon is approximately
240,000 miles from the earth.
Find the diameter of the moon in miles.
Ans. 2170 miles.
7. At what distance may a mountain 2.5 miles high be seen at sea, taking
the earth's radius at 3960 miles.
Ans. 140.7 miles.
FUNCTIONS INVOLVING MORE THAN ONE ANGLE
respectively.
Also, multiply
~~
in the same
common side of triangles DOP and LPD.
CHAPTER
FUNCTIONS
INVOLVING
VIII
MORE THAN ONE ANGLE
78. In the previous chapters, we have worked with, and
established the relations between, the functions of a single angle.
But, in solving oblique triangles and in many of the applications
of trigonometry to other subjects, formulas are used which are
derived from the functions of the sums or differences of angles.
These functions are expressed in terms of the functions of the
individual angles and are as follows for the sine and cosine:
[13]
[14]
[16]
[16]
sin
cos
sin
cos
(a +
(a +
(a (a -
~) = sin a
COS
~+
COSa sin
~.
~) = cos a cos ~ - sin a sin ~.
~) = sin a cos ~ - COS a sin ~.
~) =
COS a COS
~ + sin a sin ~.
Formulas
[13] and [14] are often called addition formulas, and
[16] and [16] subtraction
c
formulas.
79. Derivation of the formulas
for the sine and cosine of the sum
of two angles.-Lct
LAOB = a
and LBOe = (j (Fig. 84), each
of which is acute and so chosen
that a + (j = LAOe is less than
90°. In order that the functions
(j may be inA of a, (j, and a +
0
volved in the same formula, we
H
K
FIG.84.
may form right triangles which
(j
as acute angles.
have a, (j, and a +
Choose any point P in the terminal side oe.
Draw PH 1.. OA
PD 1.. OB, DK 1.. OA, and DL 1.. PH. i:::,.KOD is similar to
i:::,.LP
D, since their sides are perpendicular each to each, Then
LLPD = a.
LP
.
R) = HP = KD + LP = KD + Op'
sm (a + f.I
By d e fim' t lOn,
'
OP
OP
OP
Now multiply numerator
and denominator
of
~~
by OD, the
common side of the two triangles of which KD and OP are sides
108
way
by PD,
109
the
Then
,
sm (a + (j) = KD . OD + LP PD
OD OP PD' Op'
KD
,OD
LP
PD,
But
OD = sm a, OP = cos (j, PD = cos a, and OP = sm (j.
[13J
. . . sin (a + (j) = sin a cos (j
+ cos a sin (j.
By definition,
OK - HK
OK LD OKOD
LDPD
=
=
OP
OP-OP=OD'OP-PD'OP'
OK
OD
LD,
PD.
But
OD = cos a, OP = cos (j, PD = sm a, and OP = sm (j.
[14J
. . . COS(a + (1) = COSa COS(j - sin a sin (j.
80. Derivation of the formulas for the sine and cosine of the
difference of two angles.-Let
B
LAOB = a and LeOB = (j be
the two acute angles (Fig, 85),
c
Then angle AOe = a - {1. For
reasons similar to those given in
the preceding article, choose
~ny point P in the terminal siJ.e
OH
cos(a+{1)
oe of
(a
=OP
-
(1).
Draw
PH
1..
OA, PR 1.. OB, RD 1.. OA, and
PE 1.. DR. i:::,.DOR is similar
to i:::,.PERand LERP
= a.
By definition,
.
HP
DR - ER
sm (a - (1) =
D
H
FIG,85,
ER DR OR ER RP
OP =
OP
= OP OP= OR'OP- RP' Op'
DR
,OR
ER
RP,
But
OR = sm a, OP = cos (j, RP = cos a, and OP = sm {1.
[16]
. ' . sin (a - (1) = sin a cos {1 - COS a sin (j.
By definition,
cos (a
[16]
-
(3)
DR
A
-
OH OD + EP
OD
=OD.OR+EP,RP.
= OP =
= OP +EP
OP
OP OR OP RP OP
. . . COS
(a
-
(3)
=
COSa COS{3 + sin a sin {1,
In the proof of [16J and [16], it was assumed that a > {j.
Now
sUppose {j > a, Then a - {j
= - ({j - a).
~
PLANE AND SPHERICAL
110
By Art. 62 sin (a - (3)
By [16] -sin «(3- a)
FUNCTIONS INVOLVING
TRIGONOMETRY
sin (a
a)] = -sin «(3- a).
= sin [-«(3 COS (3 sin a).
= - (sin (3 COSa = sin a COS(3 - COS a sin (3,
=
sin (4) + 'Y) =
Substituting for the functions of 4>and 'Y
But sin (-(3)
=
4>
sin 'Y.
(180°
+
"I)
=
-sin
(3)is true when (a + (3)is an
=
cos (3.
+ 'Y) = -
(3 is put for (3. Then
= VI - sin2 a = VI - 0)2 = ~.
sin (3 = VI - cos2 (3 = VI
Substituting in the formulas,
- (la)2 = H.
sin (a + (3) = .~'la +1\ . it = }:r + H = ~g.
C(,,, (u
+ rJ)k
1:r
Example 2.-Prove
.~
that sin-I!
H
11;T'~~
+ sin-1
~.
I.H.
= tn-, using only
t he principal values of the anti-functions.
Proof.-
cos 'Y;
In the same manner it may be shown that the addition formulas
are true for any angles. It will noW be assumed that the addition
formulas for sine and cosine are true for all values of the angles.
82. Proof of the subtraction formulas for all values of the
angles.-Since
the addition formulas are true for all values of
-
(3, and cos (-(3)
rin (a + (3) = sin a cos (3 + cos a sin (3,
cos (a + (3) = COSa COS(3 - sin a sin (3.
To substitute in these formulas it is necessary first to find cos a
and sin (3.
.
(180° + 'Y)}= sin [2700+(4)+'Y)}
sin (a + (3) = sin [(90° + 4» +
sin 4>sin 'Y.
= - cos (4) + 'Y) = - cos 4>cos 'Y +
their values in terms
Substituting for the functions of 4>and 'Y
of the functions of a and (3,
(-cos a)( -sin (3)
sin (a + (3) = - (sin a)( -cos (3) +
cos
a
sin
(3.
= sin a cos (3 +
a and {3,they are true when
-sin
- sin a sin (-(3).
That is, the subtraction formulas are true in general.
Example I.-Given
sin a = ! and cos (3 = 1'\-; find sin (a + (3)
and cos (a + (3) if a and (3are acute.
Solution.-The
formulas to be used are
COSa
a sin {3+ sin a cos (3
sin a cos (3 + cos a sin (3.
COS
is true for values of the angles as given above.
(2) Suppose that a is in the second quadrant and (3in the third,
180° + 'Y. On this assumption,
such that a = 90° + 4>and (3 =
cos (90° + 4» = -sin 4>;
sin a = sin (90° + 4» = cos 4>; cos a =
= sin
+
(a - (3) = sin a cos (3 - COSa sin (3,
cos (a - (3) = COS a COS(3 + sin a sin (3.
and
That is, the formula for sin (a +
angle in the second quadrant and a and (3as stated.
ay "buw tha,t the formll1!1for ~~
sin{3
111
and
their values in terms
'Y; cos (3 = cos (180°
=
. . . sin
ONE ANGLE
(-(3)]
sin [a
= sin a cos (-(3) + cos a sin (-(3),
=
cos (a - (3) = COS[a + (-(3))
= COSa COS(-(3)
of the functions of a and (3,
sin (a + (3) =
(3)
and
which is the same result as was obtained before.
81. Proof of the addition formulas for other values of the
angles.-In
Art. 79 formulas [13] and [14] were proved when
a, (3, and a + (3are each less than 90°. They are, however, true
for all values of the angles.
(1) Suppose that a and (3 are acute and such that a = 90° - 4>
and (3 = 90° - "I, where 4> and "I are each less than 45°.cosOn
4>,
(3) > 90°, (4) + 'Y) < 90°, sin a =
this assumption, (a +
sin
'Y.
cos a = sin 4>,sin {3= cos "I, and cos {3=
+ (90° - 'Y)]
.'. sin (a + (3) = sin [(90° - 4»
(4)
+ 'Y)]
= sin [180° sin 4> cos 'Y + cos
-
MORE THAN
Then
But
Let a = sin-1 i and (3 = sin-1 t.
. . . sin a = !. and sin (3 = l
sin (a + (3) = sin a cos (3 + cos a sin (3
=.~.!+t.t=I.
sin-II = tll'.
. . . sin-1 t +
sin-1
t =
tn-.
EXERCISES
Answer Exercises 1 to 10 orally. Apply the addition and subtraction
formulas in the following expansions:
1. Expand
sin (15°
+
30°).
4. Expand
cos (23° - 10°).
2. Expand cos (45° + 15°).
5. Expand sin (240° - 30°).
3. Expand sin (75° - 60°).
6. Expand cos (30° - 120°).
7. Does sin 60° = sin (40° + 20°)?
PLANE
112
AND SPHERICAL
FUNCTIONS INVOLVING
TRIGONOMETRY
8. Does 2 sin 25° = sin (25° + 25°)?
9. Does sin (40° + 30°) = sin 40° + sin 30°?
10. Does cos (360° + 120°) = cos 120°?
(J),
11. Given sin a = H and sin {J = -l~, a and {J acute; find sin (a +
cos
(a + (J), sin
12.
Given
=
sin a
and cos (a + (J).
13. Given
-
(a
(J), and
V;, and
cos
tan
(a
{J
-
(J).
= 0,
a
Ans.
and
Hi;
{J acute;
ib;;
Hi;
tHo
(J)
find sin (a +
sin a
= !,
-
(J),
find sin (a +
and tan {J = !, a and {J acute;
(3).
(J), and cos (a
-
Ans. 0.74820; 0.66347; 0.20048; 0.97970.
14. Given sin a = -i, cos {J = -ii, a in the fourth quadrant and {Jin
(J).
the third. Find sin (a + (J), cos (a + (J), cos (a - (J), and sin (a Ans. sin (a + (J) = Hi cas (a + (J) = -at.
{Jin
16. Given cos a = -t, a in the second quadrant, and cot {J = Ji'-,
(J),
and
cos
(J),
sin
(a
(J),
cos
(a
+
the third quadrant.
Find sin (a +
Ans. sin (a + (J) = -Hi cos (a + (J) = H.
(a - (J).
16. Find sin 90° by using 90° = 60° + 30°.
17. Find sin 90° by using 90° = 150° - 60°.
18. Find cos 180° by using 180°
19. Find sin 75° by using 75°
=
+ 45°.
- 45°.
135°
= 120°
-
60°, (c) 150°
21.
Find
sin
= 75° + 75°.
120 by using
(a) 120°
= 60°
(c) 120° = 210° - 90°.
Find the values of the following cxpressions,
{)~Q1l' a!l!l;le~
22. sin (sin-cff
23. cos (sin-l
+ cos-1 H).
Js
+ tan-l~)-
24. sin (tan-l
~ + tan-l~)26. cos (tan-l i - cos-11).
26. sin [tan-l (--t) + sin-l i].
27. sin (sin-1 a + sin-1 b).
+ y'B) = 0.9659.
+ 30°,
(b) 150° = 210°
60°, (b) 120° = 90°
+
using only the principal
u
+
30°,
values
Ans. ~.
Ans.
1
0'
1
0'
Ans.0.617.
Ans. 0.05993.
Ans.
Ans. a~
+ bvT=£l2.
Ans. VI - a2V1 - b2 - ab.
Ans. VI - a2V1 - b2 - ab.
Ans. bv'!=U2 + aVI - b2.
28. cos (sin-l a + sin-l b).
29. sin (cos-1 a - sin-1 b).
30. cos (sin-1 a - cos-1 b).
Prove the following by expanding by the addition and subtraction
formulas:
34. cos (180° + 0) = -cas o.
31. sin (90° + 6) = cos o.
0)
sin
o.
36.
cos (270° - 0) = -sin 6.
32. sin (180° =
36. cas (360° - 0) = cas O.
33. sin (270° - 0) = -cos o.
2
1
37. If cos a = ...;-r/ and tan {J = 3' a and {Jare acute angles, prove that
a + {J = 45°.
113
In the following use only principal values of the angles:
12 :":.
38. Prove sin-l ~
13
39. Prove sin-l
+ sin-l 13 = 2
+ cos-1 ~ = ;.
40. Prove sin-l x + cos-1x = ;.
~
42. Prove cos-1 a :t cos-1 b
= cos-1 lab + V (1 -
a2) (1
- b2)].
cas (A
- B)
=
Find a value of 0 in the following exercises:
a)
44. cas (50° + a) cas (50°
sin (50°
+ a)
sin (50°
43. Prove that
in any right triangle
-
+
-
+
2~b.
c
-
a) = cas O.
Ans. 0 = 2a.
46. cas 30° cas (105° - a) - sin 30° sin (105° - a)
= cas O.
Ans. 0 = 135° - a.
46. sin (60° + l{J) cas (60° !(J) cos (60° + !(J) sin (60°
!(j)
= sin O.
Ans. 0 = 120°.
sin (450
47. cas (45° - x) cas (45° + x)
x) sin (450 + x)
cas O.
-
-
+
Ans.
Ans. H0
20. Find sin 150° by using (a) 150° = 120°
ONE ANGLE
41. Prove sin-l a :t sin-1 b. = sin-l (aVl - b2 :t bvI=a2).
Ans. 0.9659; -0.2588.
cos (a + (J), sin (a
MORE THAN
48. Prove that cas (a - (J) gives the same result
Prove that formulas [13] and [14] are true in the
49. a in the fourth quadrant and 13in the first.
60. a in the Ghird quadrant and {Jin the third.
61. a in the first quadrant and {Jin the third.
Expand and derive the formulas expressed in the
62. sin (a + (J + 'Y) = sin a cas 13cas 'Y + cas Cl
+ cos
Cl
cos fJ sin
'Y
-
0
=
= -2x.
whether a > {Jor a
< {J
following cases:
following:
sin 13 cas
'Y
sin Clsin (i sin 'Y.
S~.-slli--ta-+~-+-YT=-5int{<.-+
p) + 'Y].
= sin (a + (3) cas 'Y + cas (a + (J) sin
63. cos (a + f3+ 'Y) = cas a cas {J cas 'Y
sin Cl sin 13 cas 'Y
- sin a cas 13sin 'Y cas a sin 13sin 'Y.
-
'Y.
-
83. Formulas for the tangents of the sum and the difference
of two angles.-By
[7], [13], and [14],
t ana (
+ (3) -
sin (a + (3) sin a cos {3+ cos a sin {3
cos (a + (3)- cos a COS{3- sin a sin {3.
Dividing both numerator
applying [7],
tan (a + (3) =
[17]
sin a
a
cos a
COSa
COS
and denominator
cos {3
COS{3 +
COS{3
COS{3
by cos a COS{3,and
cos a sin {3
tan a + tan {3
C?S a c~s {3
sm a sm {3 = 1 - tan a tan {3
COSa COS{3
. . . tan (a + ~) = tana+tan~ .
1 - tan a tan ~
PLANE
114
AND
SPHERICAL
Similarly, tan (a - ~) = 1t~:a:
[18]
FUNCTIONS INVOLVING MORE THAN ONE ANGLE
TRIGONOMETRY
at:n~ ~.
Since formulas [13J, [14], [15], and [16] are true for all values
of exand {3,the formulas [17] and [18] are true in general. These
formulas express the tangent of the sum or of the difference of
two angles in terms of the tangents of the individual angles.
This formula may be stated as follows:
The sine of any angle is equal to twice the product of the sine and
cosine of the half angle.
Thus,
Also
Ans.3.732.
Ans. 0.268.
2. Find tan 15° by using 15° = 45° - 30°.
{jacute; find tan (a + (j)
{j
a
and
3. Given tan a = ~l.and tan = 12'
Ans. 0.8063; -0.1115.
and tan (a - (j).
{j
{j
4. Given sin a = i and cos = i, a in the second quadrant and in the
(j).
Ans.
0.4028;
1.7953.
(j)
and
tan
(a
first quadrant; find tan (a +
6. Find tan (tan-l t - tan-l +), using the principal values of the
Ans. 1.
;tngles.
6. Find tan (tan-l i - cos-1 j), using the principal values of the angles.
Ans. -0.57166.
Ans. !.
7. Find tan (tan-1 ~lr + cot-l :J-I).
Ans. f.
8. Find cot (cot-l it + cot-l J-j).
5
10. Prove that tan-l ~
+
i = !-71'.
11. Prove that tan-1 a :t tan-1 b
12. Prove
tlml
L"II-l
13. Derive formula
.
14.
Deflve
?,
+
[18].
the formula
cot (a
.
a+b
= tan-1 1 :f ab'
l - bn-' 1'"
tall-I
+
(j)
-
(j)
=
-
17. Derive the formula
)
t ana ( + {j + "1=
18. Prove that tan a
+
cos (ex + (3) = cos (ex
ex) = cos ex cos ex - sin ex sin ex
cos2
ex
sin2
ex
=
= 1 - sin2 ex - sin2 ex = 1 - 2 sin2 ex,
+
= cos2 ex - (1 - cos2 ex) = 2 cos2 ex - 1.
That is,
or
[20]
Thus,
cos 2a = cos2 a
-
sin2 a = 1
cos 30° = cos2 15°
-
-
2 sin2 a = 2 cos2 a -
sin2 15°, or 1
-
1.
2 sin2 15°, etc.
30
. 38
cos 30 = cos2 - - sm2 -.
2
2
2 tan a
.
tan 20 =
1 - tan2 a
r21]
~:r.
{j
- 1.
cot a cot
{j
co t a + co t
{j
+ 1.
cot a cot
= cot {j - cot a
(b
16. Prove that in any right triangle tan (B - A) =
16. Deflve the formula cot (a
= sin 2(3ex) = sin 6ex.
2 sin 25° cos 25° = sin 2(25°) = sin 50°.
2 sin 3ex cos 3ex
Also, tan (ex + (3) = tan (ex+ ex)= tan ex + tan ex .
1 - t an ex t an ex
That is,
71'
9. Prove that sec-1:3 - cot-l 7 = 4'
tan-l
sin 40° = 2 sin 20° cos 20°.
sin ex = 2 sin !ex cos !ex.
And, conversely,
EXERCISES
1. Find tan 75° by using 75° = 45° + 30°.
115
Thus, tan 60 ° =
1
+
aidt
tanO=1
- a).
{j tan
"I .
tan a + tan {j + tan "I - tan a tan
{j tan "I - tan "I tan a
{j
1 - tan a tan
- tan
tan cpsec a
tan (a + cp).
cos a - t' an cpSln a =
84. Functions of an angle in terms of functions of half the
angle.-Since
the formulas for the sum of two angles are true
for all values of exand {3,they are true when {3= ex.
Then, sin (ex + (3) = sin (ex + ex) = sin excos ex + cos exsin ex.
That is,
sin 2a = 2 sin a cos a.
[19]
2 tan 30°
2 tan 50°
- tan2 30°' tan 100 ° = 1 - tan2 50°'
2 tan !O
- t an 210'
"2
Example I.-Given the functions of 30°, to find the functions of
60°.
Solution.-sin
60° = 2 sin 30° cos 30° = 2(!)(hl3)
=~V3.
cos60°
= 2cos230° -1 = 2(!V3)2 -1 = t - 1 = t,
= 1 - 2 sin2 30° = 1 - 2(t)2 = !.
tan 60 °
=
or
2 tan 30°
tV3
. /0
1 - t an 2 30° = 1 - (! y- 3)2 = V 3.
Example 2.-Prove
Proof.-tan
ex
that
sin ex
= -
COSex
1
2 tan ex
.
tan2 ex = sm 2ex.
+
and 1
+ tan2
ex
=
sec2 ex.
116
PLANE AND SPHERICAL TRIGONOMETRY
2 tan a
1 tan2 a
Then
+
2
sin a
= ~sec2
2-
=
a
FUNCTIONS INVOLVING MORE THAN
Suggestion.-In
sin a
cos a
40.
1 = 2 sin a cos a = sin 2a.
cos2 a
.
the following
in terms
1. sin 4a.
of functions
of half the angle.
7. sin 20°.
2. sin !a.
8. cos 90°.
14. sin (2 sin-1 a).
3. sin 8a.
9. tan !8.
15. cos (2 cos-1 a).
4. cos 4a.
10. tan 48.
16. cos (2 sin-l a).
5. cos !a.
11. tan 80°.
17. sin (2 cos-1 a).
6. sin 90°.
12. sin 38.
18. tan (2 tan-1 a).
19. Given the functions of 75°; find sine, cosine, and tangent of 150°.
Note.-See
Exercise 19, page 112.
20. Given the functions of 150°; find sine, cosine, and tangent of 300°.
21. Given sin 8 = 07~and 8 in the first quadrant;
find sine, cosine and
tangent of 28.
Ans. 0.5376; 0.8432; 0.6376.
22. Given cos 8 = 1; find cos 28.
Ans. -0.92.
23. Given sin !8 = -f"f;and cos !8 = -H; find sin 8 and cos 8.
Ans. -0.7101; 0.7041.
24. Given tan 48 = T"O;find tan 88.
Ans. 1.0084.
Find the value of the following, using the principal values of the angles:
25. sin (2 cos-1 ~).
Ans. 0.7332.
26. cos (2 sin-1 ~).
Am. 0.68.
.
1
.
.
..1ns. 2:r
27 sm 2 sm-1
.
(
V'i +;2
28. cos (2 arc tan t).
29. tan (2 invsin t).
8 Y
)
1+ x2'
Ans. 0.8.
Am. 0.8081.
4xY(X2 - y2).
(x2 y2)2
.
2xy
.
8 = x2 + y2 and sm 28 =
30. If tan = x showthat sm
'2
+
2 cos 0
Ix - y
Y-, show tha t '\Jrx:+Y
~
+ "'J31. If tan 8
~
)(
35. Prove that
36.
Prove
that
37.
Prove
that
1
+ tan2
1
(45° - 8)
-0-- 2
(45° -8 ) = sm8
1 -ant 2
tan 2 2a
sec 2 a- 1 =
in any
right
(1
+
triangle
38. Prove that in any right triangle
.
-
4 sin3 9,
-
[13] let a
3 cos 8.
tan3 8.
sin 2A
=
sin 2B.
cos 2A
=
sin (B
(3
= 8.
=
43. cos 48
= COS4 8 - 6 COS28 sin2 8 +
-
sin4 8.
85. Functions of an angle in terms of functions of twice the
angle.-By
[20], cos 2a = 1 - 2 sin2 a. Solving this for sin a,
.
/1 - cos 2a
we have sm a= ::i:"\j~
Let a = to and we have
sin !O = + /1 - cos O.
2
-"\j
2
That is, the sine of an angle is equal to the square root of one-half of
the quantity, one minus the cosine of twice the angle.
.
. 10
/1 - cos 20°
Th us, sm 50 ° = /1 - cos 100° sm
- .
° = "\j
"\j
2
'
2
[22]
Also by [20], cos 2a = 2 cos2 a - 1.
Solving this for cos a
+ cos 2a .
"\j
2
Let a = !O and we have
we have cos a = ::i: /1
cos!O = + /1
2
-"\j
[23]
+
cos O.
2
That is, the cosine of an angle is equal to the square root of one-half
of the quantity, one plus the cosine of twice the angle.
Th us, cos 30 ° = ~1
.
tan !O = + 11
2
- "\j 1
[24]
+ cos 60°
-
cos 0
+ cos
0 =
1 - cos 0
sin 0
= 1
+ cos 100° .
2
sin 0
+ cos 0.
The last two forms given in [24] may be obtained as follows:
VI
- A).
= 28 and
41 t an 38
. .
M u It Ip Iymg numerator
sec 2a)2.
Derive the formulas given in Exercises 39 to 44.
~9. sin 38 = 3 sin 9
)
-
cos 50 ° = "\j/1
2'
By dividing [22] by [23], we can derive
~
= X
X- Y
x + Y =. v /-'cos 28
sin 2{3
32. Prove that tan {3 =
1 + cos 2{3'
33. Prove that tan 8 - cot 8 = -2 cot 28.
(3
1 - cos a + COS {3 - COS (a + (3)
34 . P rove th a t
= an « co t 21 + cos a - COS{3- COS(a + (3) (t -2
8
44 . tan 48 = 4 tan 8(1 - tan2 8).
1 - 6 tan2 8 + tan4 8
Answer
+ 2«).
13. sin (90°
COS3
117
3 tan 8
1
3 tan 2 8
42. sin 48 = 4 COS38 sin 8 - 4 cos 8 sin3 8.
EXERCISES
Express
orally.
formula
= 4
cos 38
ONE ANGLE
-
cos
tan!o =
2
.
an d d enommator
0f
1 - cos 0
b
1 + cos 0 Y
0,
/1 "\j 1
+
cos 0 . 1
cos 0 1
-
cos 0
cos 0
(1 =V
vi 1 -
COS 0)2
cos20
1 - cos 0
sin 0
FUNCTIONS INVOLVING MORE THAN ONE ANGLE
PLANE AND SPHERICAL TRIGONOMETRY
118
Again, multiplying numerator and denominator
tan!8
2
1 - cos 8 . 1 + cos 8
1+cos8
1+cos8
=
I
Th us , t an 40 ° 8
-
V(1+cos8)2
sin ~
= l+cosO
.
sin 80° .
1 - cos 80°
cos 80°
cos 80°
sin 80°
1 + cos 80°
=
Then cos 8 = !.
cos-1!.
~I -
= sin~8 =
sinGcos-l~)
VI
cos2 {
~1 +
the value of sin (! cos-1 i).
Example.-Find
Solution.-Let
-
=
cos 8,
by VI +
~OS8
=
~I;!
=
~
=
~.
the following
in terms
of functions
of twice the angle.
Answer
11. sin (45° +
sin 2a.
6. tan 90°.
sin ia.
7. tan 20.
12. cas (45° - !a).
cas 4a.
8. tan iO.
13. sin (135° +
cas 50°.
9. sin 30.
14. cas (135° - !a).
16. tan (135° - !a).
10. cas 80°.
sin 70°.
Given cas 60° = !; find sin 30° and cas 30°.
find tan 6n°.
Given cas 135° = -!V2;
18. Given cas 270° = 0; find sin 135° and cas 135°.
19. Given cas 30° = !V3; find sin 15° and cas 15°.
20. Given cas 20 = i; find sin 0 and cas o. Ans. :1:0.6325;:1:0.7746.
21. Given cas 0 = 0.6; find sin !O and cas !O. Ans. :1:0.4462; :1:0.8946.
i V':.!;
22. Given cus SO = 23. Given sin 0
Mn !O.
) . 21
(aSID"2(b)
(e)
0
-
2 =
COS2!.O
tan2!.O
find sin ,Ie and ros 4e,
Ans. :1:0.9239; :1:0.3827.
= - i, e in the third quadrant; Ans.
find sin !e, cas !O, and
0.9920; -0.1260.
b2
24. Given cas 0 =
+ e2 2be
(s -b)(s -e).
be
s(s - a).
a2
' and 2s = a
+ b + e; prove
that
.
2 =
s(s - a)
Find the value of the following, using the principal values Ans.
of the0.9923.
angles:
26. cas (! tan-1 H).
Ans. 0.4472.
26. sin G cot-l 1).
1;
,~-, x> 1.
2VX .
Ans. 0,0
< x <
vX
27 . tan !.2 sin-1 l+x
Ans. -0.14142.
28. sin (,r + 1 sin-1 27r,).
Ans. 4.8867.
sec-1
i).
29. tan (60° + l
Ans. 0.76901.
30. sin (120° + l sin-1 i).
Ans. 0.12597.
:'11. cas (90° - l sin-l t).
[
+
(
37.
(
)]
2
l
="2
) (
2
(3
(cot "2 - tan
In [24], show
why
the
)
.
2')
sm2 a
(3
sign
+ is not
- sm2 {3
necessary
before
1
-
-.smcoos 0
and
86. To express the sum and difference of two like trigonometric functions as a product.-In
this article the following
formulas are proved:
[25]
[26]
[27]
[28]
sin n + sin ~
sin n - sin ~
cos n + cos ~
cos n - COS~
=
=
=
=
2 sin !en + ~) cos !en 2 cos !en + ~) sin !en 2 cos !en + ~) cos !en -2 sin !en + ~) sin !en
~).
~).
~).
- ~).
The object of these four relations is to express sums and
differences of functions as products.
In this manner formulas
can be made suitable for logarithmic computations.
Proof of [25] and [26].--Let ex = ;t + y and p
= x - y.
Solving simultaneously for x and y,
x = !ea + (3) and y = Ha - (3).
By [13], sin a = sin (x + y)
= sin x eos y + cos x sin y.
By [15], sin {3
= sin (x - y) = sin x cos y - cos x sin y.
By adding (a) and (b), sin a
Substituting
be'
(s - b)(s - e).
~
sin 0
1 + cas 0
!a).
!a).
1.
2.
3.
4.
6.
16.
17.
-
e - b
. 1
Ie b
an d sm A = ''\/~.
2' = -a'
2'
33. Prove that tan iO and cot io are the roots of x2
- 2x csc 0 + 1 = O.
Prove the following identities:
{3
1
34. 1
tan {3tan
2' = cas {3'
a + (3
a - (3
(~in a + si.n (3)2.
36. tan
cot
=
36. cot (3
EXERCISES
Express
orally.
. t trIang
.
32 . I n any rIg
h
Ie prove tan 1A
119
{3 = 2 sin x cos y.
the values of x and y, we have
sin a
Subtracting
sin a
-
+ sin
(a)
(b)
+ sin
(3 = 2 sin
!ea + (3)cos !ea - (3).
(b) from (a) and substituting
for x and y,
sin {3 = 2 cos x sin y
= 2 cos !(a
+ (3)sin tea - (3).
Proof of [27] and [28].By [14], cos a
By [16], cos {3
= COS (x
+
y)
= cos x cos y - sin x sin y.
= cos (x - y) = cos x cos y
+ sin
x sin y.
(c)
(d)
Adding (c) and (d) and substituting for x and y,
cos a + cos {3= 2 cos x cos y
= 2 cos !(a + (3) cos !(a - (3).
120
PLANE
AND
SPHERICAL
FUNCTIONS INVOLVING MORE THAN ONE ANGLE
TRIGONOMETRY
Subtracting (d) from (c) and substituting for x and y,
(3)sin Ho: - (3).
cos 0: - COS (3 = -2 sin x sin y = -2 sin Ho: +
cos 70° - cos 30°
as a product.
Example I.-Express
cos 70° + cos 30°
Solution.By [28], cos 70° - cos 30° =
=
By [27], cos 70° + cos 30° =
=
Then
-2 sin H70° + 30°) sin H70
-2 sin 50° sin 20°.
2 COilH70° + 30°) cos !(70°
2 cos 50° cos 20°.
-
30°)
-
30°)
cos 70° - cos 30°
- 2 sin 50° sin 20°
= 2 cos 50° cos 20° =
cos 70° + cos 30°
-
tan 50° tan 20°.
Example 2.-Show that the following equality is true by using
the tables to compute each side of the equality:
sin 60° + sin 40° = 2 sin 50° cos 10°.
Solution.-The
right-hand
member is best computed by
logarithms.
x = 2 sin 50° cos 10°
sin 60° = 0.8660 Let
log
2 = 0.30103
sin 40° = 0.6428
log
sin
50°
= 9.88425
sin 60° + sin 40° = 1.5088
log cos 10° = 9.99335
fO~1:-
=QJ786Q_-
u
The two results are found to agree.
Example 3.-If 0: + {3+ 'Y = 180°, prove the identity:
sin 0: + sin {3+ sin 'Y = 4 cos !o: cos !{3 cos h.
Ho:
Ho: + (3)2cos to: cos !{3
(0: + (3)cos !o: cos !(3.
EXERCISES
and differences
sums
of functions
as products.
7. sin 3£1 + sin £I.
.
25.
26.
27.
+ (3)]
= 2 sin
= 4 sin!
2. sin 70° - sin 50°.
8. cos 5£1- cos 7£1.
3. cos 70° + cos 50°.
9. cos 7£1- sin 3£1.
4. cos 70° - cos 50°.
10. cos 2ex - cos 2{3.
5. sin 80° + sin 140°.
11. sin (ex + (3) + sin (ex - (3).
6. cos 140° - cos 70°.
12. cos (ex + (3) - cos (ex (3).
Express the following as products and simplify:
13. sin 80° - sin 40°.
Ans. sin 20°.
14. cos 80° - cos 40°.
Am. -0
sin 20°.
15. cos 40° + cos 20°.
Am. 0 cos 10°.
16. sin 40° + sin 20°.
Ans. cos 10°.
17. sin 50° + sin 70°.
Am. 0 cos 10°.
sin 50° - sin 30°.
18
Ans. tan 10°.
cos 50° + cos 30°
sin 70° + sin 50°.
19.
Ans. -cot 10°.
cos 70° - cos 50°
cos 50° + cos 10°.
20 .
A ns. cot 30°,
sin 50° + sin 10°
cus-,y-=-=-rr
.' {3
sm ex sm
u
Am.
+
cos 2£1+ cos £I.
sin 2£1+
28.
+
13).
H'" -
-
ex).
cos (ex
30°).
sin (ex - 60°).
cot ({3 + 171").
sin (ex + 60°) +
tan ({3 - 171") +
Solve cos 3£1 + sin 2£1
Solve
---------------------------------
-tan
A ns. c ot Ji£l
2'
sin £I
24. cos (ex + 30°)
(3) (Art. 48).
.'. sin'Y = sin [180° - (0: + (3)] = sin (0: +
By [19], sin (0: + (3) = 2 sin !(o: + (3)cos Ho: + (3).
. . . sin 0: + sin (3 + sin 'Y
(3) cos Ho: - (3) + 2 sin Ho: + (3) cos Ho: + (3)
= 2 sin Ho: +
(3) + cos Ho: + (3)].
= 2 sin Ho: + (3)[cos Ho: -
-
'Y
23. cos (60° + ex) + cos (60°
By [25], sin 0: + sin (3 = 2 sin Ho: + (3)cos Ho: - (3).
Now 'Y = 180° - (0: + (3).
(3)
Express the following
Answer orally.
1. sin 70° + sin 50°.
22 .
Proof.-
-
sin
But Ho: + (3) = 90° - h, and sin Ho: + (3) = cos h.
.'. sin ex + sin {3+ sin 'Y = 4 cos !o: cos !{3 cos h.
21..
x = 1.5088
But cos Ho: - (3) + cos ! (0:+ (3)
(3)]cos ![!(o:
= 2 cos ![!(o: - (3) + Ho: +
= 2 cos !o: cos !{3.
. . sin 0: + sin {3+
121
-
-
cos £I
sin 3£1 + sin 2£1 + sin
£I
=
=
Ans. cos ex.
Am. V3 cos ex.
Am.
0 for values
sin ex.
Ans. O.
of £I < 360°.
Ans. 0°, 30°, 90°, 150°, 180°, 270°.
0 for values
of £I < 360°.
Am. 0°, 90°, 120°, 180°, 240°, 270°.
29. Solve cos 3£1 - sin 2£1 + cos £I = 0 for values of £I < 360°.
Ans. 30°, 90°, 150°, 270°.
30. Solve sin 5£1 - sin 3£1 + sin £I 0 for values of £I < 360°.
=
Ans. 0°, 30°, 60°, 120°, 150°, 180°, 210°, 240°, 300° 330°.
If ex + {3 + 'Y = 180°, prove the identities in the following exercises:
31. sin ex + sin {3 - sin
4 sin !ex sin !{3 cos h.
'Y =
82. cos ex + cos {3+ cos 'Y = 4 sin !ex sin !{3sin h + 1.
33. cos 2", + cos 2{3+ cos 2'Y= -4 cos excos {3cos 'Y - 1.
34. sin 2", + sin 2{3+ sin 2'Y = 4 sin exsin (3sin 'Y.
FUNCTIONS
122
PLANE AND SPHERICAL TRIGONOMETRY
=t
87. To change the product of functions of angles into the
sum of functions.-From
Art. 78,
(a)
sin (a + (3) = sin a cos (3 + cos a sin (3.
sin (a - (3) = sin a cos (3 - cos a sin (3.
(b)
(c)
cos (a + (3) = COS a COS (3 - sin a sin (3.
(d)
COS(a - (3) = COS a COS (3+ sin a sin (3.
(3) + sin (a - (3) = 2 sin a cos (3.
Adding (a) and (b), sin (a +
sin (n - ~).
[29)
. . . sin n cos ~ = ! sin (n + ~) + !
Subtracting
(b) from (a),
sin (a
+ (3) -
sin (a
-
. . . cos n sin ~ = ! sin (n
[30)
Adding (c) and (d), cos (a
[31)
.'.
COS n COS
(3)
2 cos a sin (3.
+ ~) -
+ (3)+
~ = ! cos (n
=
! sin (n - ~).
cos (a
+
~)
-
(3) = 2 cos a
+ ! cos
(3.
COS
cos (a
[32)
+
-
.'. sin n sin ~ = -!
(a
-
(3)
cos (n
+
= -2 sin a
!
"
sin 60
+!
~)
+!
+ ! sin
20.
sin 0 cos 0 = ! sin 20.
Therefore,
sin2 0 cos 0
sin 0 (sin 0 cos 0)
= ! sin 20 sin 0
By [32},
= ![ -! cos (20
0)
+!
cos (20
(
)
cos 40 cos 0)
+!
cos 30)
13
cos
g".
13 +
+ cos
-
</>
cos
37r
13 +
3</>cos </>
-
O)}
57r
cos2 </>=
O.
O.
13 =
cos
14. sin 160° sin 120° sin 80° sin 40° = ."..
16. cos 160° cos 120° cos 80° cos 40° = .,(..
16. cos2 </>sin2 </>=
HI - cos 4</».
-
sin 6</».
'i(cos </>- ! cos 3</>- ! cos 5</».
19. "in' </>= 1.(3 sin </>- sin 3</».
COS3 </>= 1(3 cos </>+ cos 3</».
sinm </>cosn </>= ;~(2
cos 2</>- 2 cos 4</>+ cos 6</», if m
= 4, n
22. Work out other combinations
of m and n as illustrated in [21].
20.
21.
-
= 2.
88. Important trigonometric series.-The
trigonometric series
given in this article and the following exercises are important,
especially in certain problems in electricity.
In the series, a
and {3are angles and n is an integer equal to the number of terms
in a series. The two fundamental series with their sums are:
(1) sin a
cos 30 + t cos O.
5 cos30 +cos50).
Example 3.-Prove that coss 0 = -h(10 cos 0 + 2
1 + cos 20 cos 0 By [23).
2
Proof.-coss
0 = (cos2 0)2 COS0 =
cos2 20) COS()
= HI + 2 cos 20 +
= -
t
+
7r
18. sin2 </>cos3 </>=
sin 20.
Then
+
Apply the formulas of this article to the following.
Answer orally.
1. sin 50° sin 40°.
6. sin 6</>cos 4</>.
2. sin 40° cos 20°.
6. cos 4</>sin 8</>.
3. cos 20° sin 50°.
7. sin 20</>sin 12</>.
4. sin 70° cos 10°.
8. cos 1O</>cos 16</>.
Prove the following identities:
9. 4 cos 2</>cos 4</>cos 6</>= 1 + cos 4</>+ cos 8</> + cos 12</>.
10. 4 sin 2</>sin 4</>sin 6</>= sin 4</>+ sin 8</>- sin 12</>.
sin 6</>cos 3</>- sin 8</>cos </>
11
t an 2 </>.
sin 4</>sin 3</>- cos 2</>cos </>=
17. sin3 </>cos3 </>= ;~(3 sin 2</>
It is often desirable to express the products and powers of sines
and cosines as sums of functions that involve multiples of the
angle. The formulas of this article and of Art. 84 can be used
for this purpose (see also Art. 127).
Example 2.-Prove
that sin2 0 cos 0 = -t cos 30 + t cos e.
Proof.-By
[19}, sin 20 = 2 sin 0 cos O.
=
4 cos 20 cos 0
EXERCISES
13. 2 cos
sin (3.
cos (n - ~).
sin 60
"2
+
123
0 By [23}.
= H3 cos 0 + 2 cos 30 + 2 cos 0 + ! cos 50
= n(lO cos 0 + 5 cos 30 + cos 50).
+ sin2
Example 1.-Prove that sin 40 cos 20 = !
lying 1291,where a = 40 and (3 = 20,
- ?e
=
= H3 cos 0
12. sin 3</>sin </>
COS
)
2 cos 20
ONE ANGLE
.
(n - ~).
Subtracting (d) from (c),
(3)
= H3
THAN
1 + cos 40
cos
+
2
+ 4 cos 20 + cos 40) cos 0
(+
1
INVOLVI.\'G MORE
+ sin (a
+ (3) + sin (a + 2(3) + . . . sin [a + (n - 1){3]
sin [a + !en - 1)f3]sin !nf3.
sin !{3
(2) COSa + cos (a + (3)
+ cos (a + 2(3) +.. . COS[a + (n - 1)f3]
cos [a + !en - 1){3]sin !nf3.
sin !{3
In both (1) and (2) sin !{3 ~ O.
Proof of (1).-Let Sn = the sum of n terms of the series.
-
PLANE AND SPHERICAL TRIGONOMETRY
FUNCTIONS INVOLVING MORE THAN ONE ANGLE
Multiplying each term by 2 sin !13 and applying [32] to each
product, we have the following equations, one equation resulting
from each term:
GENERAL EXERCISES
Prove the following identities by transforming the first member into the
second:
1. cos 3060 + cos 2340 + cos 1620
+ cos 180 = O.
tan x + tan y sin (x + y).
2
tan x - tan y -- sin (x - y)
1 - tan x tan y
cos (x + y).
3.
1 + tan x tan y = cos (x - y)
1 :I: tan y.
4. tan (450 -y
+ ) =
l+tany
5 . cot (450 + ) = cot y + 1.
- Y
cot Y :I: 1
(x + y)
6. cot x :I: tan y = cos
.
.
sm x cos y
. sin 1100 cosv'3100
7---=4
.
124
2 sin a sin !13 = cos (a
-
-
!(3)
cos (a
cos (a
+ !(3).
+ !(3).
.
2 sin (a + (3) sin !13 = cos (a + !(3) 2 sin (a + (3) sin !13 = cos (a + !(3) - cos (a + 1-(3).
, ......................................
2 sin [a
+
-
(n
1)13] sin !13
=
cos
(a + 2n ;- 3(3) cos
2n
( +~
a
-
1
13
).
Adding these and noting that the sum of the first members is
Sn
.2
sin !13,
Sn
. 2 sin !13= cos (a - !(3) - cos a + -r13
(
2n
8. i[cot
1
).
sin !{J
which is true if sin !{J ~ o.
The proof of (2) can be carried oul in an exactly similar manner
multiplying by 2 sin !{J and applying [30].
EXERCISES
Suggestion.-In
2a
+ sin
3a
+ . . . sin
na
=
(1) put {3= a.
2. cos a + cos 2a + cos 3a + . . . cos na =
sin ia
sin !na .
cos Hn +. l)a
1
Sin .a
.
SI~2 na.
3. sin a + sin 3a + sin 5a + . . . sin (2n - l)a =
4. cos a
.
5 .sma
+
cos 3a
+
.
+ sma( + n )
2,,"
~
(
[
)
= O.
6. cosa + cos(a + 2:) + cos(a + :)
= o.
Sin a
sin 2na
+ . . . cos (2n - l)a = 2 Sin. a .
2(n - 1)""
4...
.
.
n
+ sma + n + ...sma+
J
cos 5a
+
. . . cos[a +
2(n
tan 0] = cot 20.
2 = tan 2 a +
+
cot a -
:
I),,"
J
cot fJ
= -
tan
a tan fJ.
15. tan (450 + 0) - tan (450 - 0) 2 tan 20.
=
16. see' \t7T"+ !a)
see a.
2tall(I"" + !a) =
sec2 ! ~ + a
t an
18.
( ) ~.
=
(,,"+1
4 2a)
2 2
17.
Prove the following, where, in each, the denominator of the sum must be
different from zero:
sin !en + l)a sin ina .
+ sin
-
COP
a.
(2 - sec2 a)(2 - CSC2a)
10.
= -cos22a.
sec2 a csc2 a
11. 2a + 4 tan-l (see a - tan a)
= "".
12. -2a + 4 cot-l (sec a + tan a) = "".
tan a + tan tI
13.
cot a + cot fJ = tan a tan {3.
tan a - tan fJ
14.
Applying [28] to the second member of this,
Sn . 2 sin!{J = 2 sin [a + Hn - 1)13sin !n{J.
sin [a + Hn - 1){J]sin !n{J,
. . . Sn =
1. sin a
0
9. 4 cot2 2a
COSa
-tan 0
2 cos 0
1
19.
1
1
+ 41- (1 + tan2 -0
2 )cot
+ s!n 0 - cos 0
+ sm 0 + cos 0 = tan 2~o.
20. 2 s!n 0
-
s!n 20
2
2 sm.O
tan2 ~O.
2
2 sm 0 + sm 20 =
21. 8 sin' !O
8 sin2 !O
-
1 = ---'-'
sec' 0
-0
+ 1 = cos
20.
22. sin (a + (3)sin (a - fJ) sin2a
=
- sin2{3 cos2fJ - cos2a.
23. cos (a + fJ) cos (a - fJ) cos2 a - sin2 =fJ cos2 fJ - sin2 a.
=
=
24. cos (a - fJ) = 1 + tan a tan fJ.
cos (a + fJ) 1 - tan a tan fJ
4 sin a sin (600 a)
.
25 .
+
-- sm 3 a.
csc (600 - a)
sin a - vI + sin 2a
26 .
co a.
cOSa - vI + sin 2« = - t
125
126
PLANE
AND
SPHERICAL
sec2 9
8.
3 sec2 9
3 tan 9 + 1 tan 9 + 3 = 3 + 5 sin 29
In the two following exercises a, {3,and -yare the angles of any triangle:
28. sin a cos {3+ cos a sin {3= sin -y.
29. tan a + tan {3+ tan -y = tan a tan (3tan -y.
cos 69 + 6 cos 49 + 15 cos 29 + 10
30 .
= 2 cos 9.
27.
+
cos 59
5 cos 39
+
10 cos 9
Suggestion.-Write
the numerator
+ 5 cas 29 + 10 cos 29 + 10.
Apply
[27] and
in form cas 69 + cas 49 + 5 cos 49
[23], and this becomes
2 cos 59 cas 9 + 10 cas 39 cos 9
+ 20 cos2 9 = 2 cos 9(cos 59 + 5 cos 39 + 10 cos 9).
31. sin 59
=
+ 5 sin 9.
+ 5 cos 9.
!.
39. tan-l
.-l
40 . t all
1
1
+a+
tan-l
G) +
-1 -1
a
+
2y - x
t. . -I .,'11t
'. - -
y
2x -
--r;=yv
tan-l
+
x v;}
3
2 tan-l
tan-l
G) =~.
2
-a2 = n."..
r;<
~
tall-l
-----------
Solve the following equations for values of the angle less than 360°:
Ans. 0°, 120°, 180°, 240°.
41. sin 29 + 2 sin !9 cas !9 = O.
Ans. 0°, 60°, 180°,300°.
42. sin 9 cos 9 - sin !9 cas !9 = O.
Ans. 90°, 210°, 270°, 330°.
43. sin 29 + coso !9 - sin2 !9 = O.
Ans. 22!0, 67!°, 112!0, etc.
44. sin2 29 - cos2 29 = O.
45. cas 29 - sin2 9 = O.
Ans. 35° 15.9'; 144° 44.1'; 215° 15.9'; 324° 44.1'.
46. sin 29
47. sin 49
+ cos
+ sin
29
29
+ sin
+ cos
9 = 1.
9
Ans.
0°, 65° 42.3',
180°, 204° 17.7'.
= O.
49.
50.
51.
52.
53.
54.
tan
+
(i.".
Ans. 60°.
tan (80° - !9) = cot j9.
Ans.
33jO,
153jO,
273jo.
9)
cot (40° +
= tan !9.
3) cot 2x.
(tan x sin x) (tan x - sin x) - cos2 x = 2(sec2 x Ans. 116° 33.9', 296° 33.9'.
Ans. 53° 7.8', 90°, 180°, 270°.
1 - sin2 x cos x = sin 2x.
Ans. 60°, 120°, 240°, 300°.
(4 cas" 9 + 1) tan2 9 = 6.
.
cos 29
+
=sm 2 9 .
tan 9 - 1 + tan (9 - 45°) - C
~
+
+
)
Ans. 60°, 120°, 240°. 300°.
ONE ANGLE
127
+ IH."..
9)
+ I)IV;
(2n
+
+
Ans.
IH.". :t 1."..
n.". :t i1I".
61. sin-1 2x - sin-l V3x
= sin-l x.
Solution.- Taking the sine of both members of the equation,
-
v'1=4X2 . V3x = x.
x(2~
- V3-v'l=4X2
0,
2vT=3X2
- V3-vT=4X2
=
Transposing and factoring,
Equating each factor to 0, x
Solving these equations,
x
= 0, and x = :t!.
All of these values satisfy the equation when principal
are used.
.
62. tan (cos-1 x)
63..sin-l
= sm
(cot-l:21) .
Cx2) = ;.
64. tan-l 2x tan-l 3x = i."..
65. tan-l x
2 cot-l x = 135°.
.
1
-x
66. sm -.". - 2 tan - 1
68.
69.
70.
71.
~+ )
-1
X2
x
Ans.0; 13.
Ans. 1.
Ans. 1.
= a.
A ns. a.
2
2
('()~-l
t:m-1
-:--'
= 3-71'.
+
I
X'
3-"--=t.
" - 1
sin':"l '(1x) + sin-l (!x)
= i."..
cot-l (x - 1) - cot-l (x
+ 1) = i..."..
cot-l (x) - cot-l (x + 2)
= 15°.
Given a sin 9
b cos 9
= e, and a cos 9
u
.. '
{i
..
.
~
X =
a cos
<pfrom the following
.
<p, y
b sm
=
<pfrom
._1n".- V/ii3.
-
---
.
-
+
72. Eliminate
1) = O.
- 1 = O.
Ans. v'5
3'
+
+
(2
(
-
values of the angles
G) + sin-1
73. Eliminate
Ans. 70°, 90°, 110°, 190°, 230°, 270°, 310°, 350°.
Ans. 10°, 130°, 250°, 330°.
48. 9 = sin-l (cos 29) - 60°.
THAN
Ans. (2n
tan (i.". - 9) = 4.
Solve the following equations for x:
60.
67.
\.- 3.
---
MORE
59. cos 79 + cos 59 + cos 39 = O.
2xvT=3X2
33. tan {sin-l [cos (tan-l x)]} = x
3a - iz"
34. 3 tan-l a = tan-l_. 1 - 3a2
1
1
tan-l 1
tan-l
35. tan-l
2a2
1 + 2a + 4a" =
1 - 2a + 4a2 +
1 tan-l -1 + tan-l 1- + tan-l -1.".= -.
36. tan-l 35784
- +
bx
bx
.
37. tan-l
= sin-l Va2 - X2~2
ava2 - b2 - x2
(0.2)
INVOLVING
Solve the following equations, giving the values in general measure:
55. sin 29 + 1 = tan (9 + 45°).
Ans. n.". + £"";n."..
56. cos 59 + cos 39 + cos 8 = O.
Ans. i(2n + 1).".;!C3n :t 1)."..
57. cos 79 - cos 9 = O.
Ans. in.".; jn."..
58. sin 59 + sin 39 = O.
Ans. in.".;(2n
16 sin' 9 - 20 sin" 9
32. cos 59 = 16 coso 9 - 20 cos" 9
38. 2 tan-l
FUNCTIONS
TRIGONOMETRY
Ans. V1.022.
:t (1 + V3).
Ans. V3, -V3 - 2.
An".
-
b sin 9
Ans.
equations:
a2
d; eliminate
+= b2 =
x2
<p.
Ans.
the following
(i2
equations:
9.
+ d2.
e2
y2
+ b2 =
1.
a cas <p + b sin <p e, b cos <p
=
+ e sin <p = a.
(e2
Ans. (be - a2)2
+
- ab)2 = (ae - b2)2.
Suggestion.-Solve for sin <pand cas <P,then square and add.
74. Given P cas 9 - W sin a
= 0 and R + P sin 9 - W COSa = 0;
solve for R.
W COS(a
9).
Ans. R =
+
75. Eliminate
csc 9 -
9 from
the following
sin 9 = a, sec 9 -
Suggestion.-From
Find aibi (oi + bi).
the
first
cas 0
a
cos 9
equations:
= b.
C~S2 O.
= smO
Ans.
From
aibi(ai
the
+ bi)
second
b
= 1.
sin2 0
=
cosB
76. Eliminate
(Jand
'P
+
-
Am.
a2
+ b2 -
-
sin2 (J
(12
Y cos (J =
2c
= 2.
+
cos2 (J
1
/)2
= x' +
y'
1.
Am. x'
U2 + II' =
y"
the first equation and collect the terms in x' and y'.
Suggestion.-Square
This gives the square
_::.
Then tan (J =
y
(J + y sin (J = O.
of x cos
From
this find sin (Jand cos (Jand substitute in the second equation.
1
2 sin a sin {3
.
((J (3)
= tan' (J,(J = tan-l sin (a (3)
78. Show that If tan ((J - a) tan
"2
Suggestion.-Write
+
the equation in the form
sin ((J
-
a) sin ((J
-
(3)
sin' (J.
cos ((J- a)cOS ((J- (3)- cos' (J
Applying
[32] and [31],
-
(J
! COS (a (3) sin'
(2(J - a - (3) + ! COS(a - (3) = cos' O'
! cos
(3) +
-~ cos (2(J - a Clearing
of fractions
and uniting,
or by composition
and division,
cos
Applymg
[16],
cos 2(J COS (a
Then
.
Applymg
[28],
+ (3) +
(a
cos (a'-.{3) tan 2(J =
sm (a + (3)
2sinasin{3 .
tan
79. Given aI' cos
that anI'
~.
= c cos
2(J
(J
=
+ bI
COS
+ (3)
(J
= c(J, and
+ (3).
-
cos
an2I2
-
~ (J; prove
bnI = c cos
and add the second.
the first by ~
cos (J
80. Show that if tan 'P =
(J
(3).
sin (a
(J
Suggestion.-Multiply
A sin
sin 2(J sin (a + (3) = cos 2(J COS(a
~,
B2 sin
A2
(J
+ B cos = Y +
((J
+ tan-1 ~}
(J W sin (J;show that
81. Given I = W sin (J,and P cos =
1
1
1
]2 = p2 + W2'
=
ONE ANGLE
Am. x = a cos
y = a sin
(J
(J
sin (J.
-+ bb cos
(J.
(J
( - ",y 4LC +- RC",
R2C2)
2 cot
129
for x and y:
k sin (J
83. Show that.
sm cot-l
v'X'+Y'.
MORE THAN
= b.
(Jfrom the following equations:
x sin (J
INVOLVING
82. Solve the following equations
x cos (J + y sin (J = a.
x sin (J - y cos (J
from the following equations:
sin (J + sin 'P = a.
cos (J cos 'P = b.
cos ((J
'P) = c.
77. Eliminate
FUNCTIONS
AND SPHERICAL TRIGONOMETRY
PLANE
128
2k
y(LC",2 -1) sin'(J + !RC",sin2(J + 1.
4LC
- R2C'
"'v
OBLIQUE
First Proof.-In
the perpendicular
of the triangles
TRIANGLES
131
Fig. 86, let ABC be any triangle, and let h be
from B to AC.
The following applies to each
(a), (b), and (c); but note that in triangle (c)
h = a.
CHAPTER
IX
OBLIQUE TRIANGLES
CASE IV. Given the three sides.
Since there are six parts to a triangle, and, in each of the four
cases, three parts are given, then, in general, there are three
unknown
parts to be found in solving a triangle.
Also, since
Jl
~
pB"~h
'..ex,
A
c
i;---
~.
~
CAb
l
of the opposite
angles.
130
= h-.
c
h
(2)
sm 'Y = (i'
Dividing (1) by (2), there results
C
sin a
a
a
c
;-= - or ;-- = -:-'
sm
'Y c'
sm a
sm 'Y
Similarly, drawing perpendiculars from A to CB,
sin (3 b
b
c
(4)
or
=
sin
(3
=
sin
'Y'
sin'Y
C'
Hence, uniting (3) and (4), there
results
a
b
c
[33] . sin
- a = sin~
= -.siny
(3)
Second Proof.-In
Fig. 87, let ABC
be any triangle.
About the triangle
circumscribe a circle. Let 0 be the
center. Draw the radii OA, OB, and
ac.
Dl'a'X OD pC'.l'pcndicular to..4.CrThen LAOD = (3or is the supplement
In triangle AOD,
(c)
three independent
equations are necessary and sufficient to
determine three unknowns, it is necessary to have three independent formulas or relations connecting the parts of a triangle.
These three relations are:
(1) The sum of the angles of a triangle is equal to 180°.
(2) The sine theorem, or the law of sines.
(3) The cosine theorem, or the law of cosines.
For greater convenience in carrying out the numerical work of
the solutions, various other relations are derived from the formulas growing out of the sine theorem and cosine theorem.
90. Law of sines.-In
any triangle
the sides are proportional
to the sines
SIn a
.
89. General statement.-In
the present chapter methods for
solving any triangle will be developed.
As pointed out in
Art. 38, it is possible to solve a triangle whenever there are
enough parts given so that the triangle can be constructed.
The
constructions and, likewise, the solutions fall under four cases,
depending upon the parts given and required:
CASE I. Given one side and two angles.
CASE II.
Given two sides and an angle opposite one of them.
CASE III.
Given two sides and the included angle.
B
.
(1)
AD
FIG. 'K7.
of (3.
= AO sin LAOD.
. . . tb = R sin (3.
In a similar manner,
tc = R sin 'Y,
and
ta = R sin a.
a
b
c
These give
sin a = sin (3 = sin 'Y'
COROLLARY.-The constant ratio of a side of the triangle to the
sine of the opposite angle is equal to the diameter of the circumscribed
circle.
.
lD
.
enve
t h e proportIOn
'
2. Derive 2R = ~,also
sma
EXERCISES
b
c
-;--;;
-'--'
sIn... = sm
'Y
2R = ~.
sm'Y
132
3. Solve sma
~
=~
sm
{3
91. Law of cosines.-In
any triangle the square of a side
equals the sum of the squares of the other sides minus twice the product
of these sides by the cosine of their included angle.
Proof.-In
each triangle of Fig. 86,
a2
h2
=
c2
-
=
h2
+ DC2.
AD2 and DC2 = (b - AD)2.
(Notice that in (a) AD is positive, in (b) negative, and in (c)
DC is zero because D falls on C.)
. . . a2 = c2 - AD2
=
c2 - AD2
= c2 +
b2
-
+ (b + b2 -
AD)2
2b . AD
+ AD2
2b . AD.
But AD = c cos ex,
.' . as = bS + cS - 2bc cos 0:.
By similar proofs or by cyclic changes we have,
[34s]
bS = as + CS - 2ac cos ~.
c2
[34sJ
= a2 + b2 - 2ab CDS y.
[341]
The cyclic changes of letters is carried out as follows:
a changes to b.
exchanges to {3.
{3changes to 'Y.
'Y changes to ex.
b changes to c.
c changes to a.
EXERCISES
1. Are the formulas [341], [34s], and [34s] adapted to solving by logarithms?
2. Derive [34s] and [34s] independently.
3. Solve each of the three formulas for the angles in terms of the sides.
4. Solve a2 = b2 + c2 - 2bc cos a for b.
Ans. b = c cos a :f: va2 - C2sin2 a.
6. What does the law of cosines become when one of the angles, say 'Y,
is a right angle?
92. Case I. The solution of a triangle when one side and two
angles are given.-In
this case, it is evident that the third
angle can always be found from the equation
a + {3+
'Y
133
The sides can then be found by using the relations stated in the
law of sines, namely,
for each part involved.
4. What does the law of sines become when one of the angles, say 'Y, is a
right angle?
But
OBLIQUE TRIANGLES
PLANE AND SPHERICAL TRIGONOMETRY
= 180°.
a
b
a
c
b
c
-sin ex = -,
- ex = sin
and = --- .
sin {3 sin
'Y'
sin (3 sin 'Y
In each of these there are four parts of the triangle involved;
therefore, if any three of these parts are known, the fourth can be
found. That is, anyone of these equations can be solved for
anyone of the four parts.
Any formula not used in the solution of a triangle may be used
in checking the work. One should be certain, however, that the
check formula was not involved in the formulas used in solving.
For instance, when two equations from the law of sines have been
used to find the parts of a triangle, the third equation from the
law of sines cannot be used as a check, since the first two equations
involve the third.
Two particularly
convenient equations for checking the
accuracy of the numerical solutions of triangles are the following,
known as Mollweide's equations, from the German astronomer
Karl Brandon Mollweide (1774-1825), though why they should
bear his name is not clear, since they were known long before his
time, and were used by Newton and others.
-c- b =
(1)
a
(2)
a+ b
c- -
sin .Hex- (3)
cos ~ 1
cos
!(ex
- (3).
sin h
The certainty of these equations as a check lies in the fact
that each contains all six parts of a triangle.
Mollweide's equations are readily derived from the law of sines.
Derivation of (1).-From
the law of sines,
a =
c ~in ex,
sm
b =
'Y
c sin ex
Then
- b
a
c
sm
'Y
c sin {3
SIn-Y - srn:y
c
=
sin ex sin
sin {3
'Y
2 cos !(ex + (3)sin !(a - (3).
2 sin h cos h
= sin ![180° - (ex + (3)] = sin [90° - !(ex + (3)]
= cos !(ex + (3).
By [19], [26],
Now sin h
=
c ~in f3.
--
l
PLANE AND SPHERICAL TRIGONOMETRY
134
OBLIQUE
a
c
sin a = sin 'Y
- b - sin Ha - fJ).
. . . a--ccos h
Equation (2) can be derived in a very similar manner.
The same suggestions as were given in Art. 41 for the solution
of right triangles should be carried out here. Draw the triangle,
state the formulas, make out a careful scheme for all the
work, and, lastly, fill in the numerical part by the use of the
Tables.
Remember that in computations
time and accuracy
are of very great importance.
Time will be saved by carefully
planning the arrangement of the work. Accuracy can be secured
by checking the work at every step. Verify at every step the
additions, subtractions, multiplications,
and divisions.
Check
interpolations when using Tables, by repeating the work at each
step.
From geometry, the area of a triangle equals one-half the
product of the base and altitude.
Using b for base, h for altitude,
.
b sin 'Y.
and K for area, K = !bh. But h = c sm a, and c = SIn
;-- {3
... K =
[35]
b2 sin a sin
"(
2sin~'
Since any side of the triangle can be used as base, or the given
side, two other forms for [35] can be found. These may be
written from the formula given by making the cyclic changes in
the parts of the triangle.
Example.-Given
a = 53° 23.7', 'Y = 75° 46.3', and a = 27.64;
find {3,b, and c.
Construction
Solution.
a
Given
= 53° 23.7'.
75° 46.3'.
'Y
- 27.64.
{ a ::
fJ
To find * b
= 50° 50'.
:
26.695.
{ c - 33.375.
b
Formulas
a
+ fJ+
'Y
=
180°
a
b
sin a - sin fJ
. Values to be put in after solving.
TRIANGLES
... fJ-- 180° - (a
a sin fJ
.'.b=
sina'
+
'Y).
. . .c=
135
a sin
sina'
'Y
Logarithmic formulas
+ log sin fJ + colog sin a.
log b = log a
log c = log a
+ log
sin
'Y
+
colog sin a.
Computation
fJ
=
log
log sin
colog sin
log
180° -
a =
{3 =
a =
b =
b =
(53° 23.7'
+
75° 46.3')
1.44154
9.88948
0.09541
1.42643
26.695
-= 50° 50'.
log a = 1.44154
log sin 'Y =
colog sin a =
log c =
c =
9.98647
0.09541
1.52342
33.375
Check by Mollweide's equation:
c
-
a
b
=
s'n .!('Y
2
- fJ) , or c
cos !a
- b=
a sin .!('Y
2
- fJ) ,
cos !a
log a = 1.44154
Note.' Use Mollweide's equation with the middle-sized side in the
denominator.
a = 27.64
log sin H'Y - fJ) = 9.33426
cologcos !a = 0.04896
H'Y - fJ) = 12° 28.1'
log (c - b) = 0.82476
!a = 26° 41.8'
c - b = 6.680
c - b = 6.680
EXERCISES
1. Given fJ, 1', and a; to find a, b, and c. Give formulas and scheme for
solution.
2. Give the formula for area when b is the given side. When c is the
given side.
3. Given a = 40° 5.5', fJ = 28° 34.4', c = 267.95;
find a = 185.26, b = 137.58, l' = 111° 20.1'.
4. Given a = 58° 9', fJ = 41 ° 41.2', c = 108.85;
find a = 93.84, b = 73.472, l' = 80° 9.8'.
6. Given a = 23° 4' 8", l' = 33° 9' 22", c = 5.94;
find a = 4.256, b = 9.028, fJ = 123° 46' 30", K
= 5.265.
6. Given fJ = 34° 47.3', l' = 109° 26.3', a
= 322.4;
find b = 314.66, c = 520.09, a
= 35° 46.4', K = 47,833.
7. Given fJ = 56° 21.3', l' = 55° 17' 37", b 89.042;
=
find a = 99.42, c = 87.93, a
= 68° 21.1', K = 3,638.7.
8. Given a = 144° 8.4', fJ = 25° 19.2', b 430.10;
=
find a = 589.14, c = 183.96,
l' = 10° 32.4', K = 23,174.
Solve the following and check by Mollweide's equations:
9. Given a = 47° 16.2', fJ = 75° 41.4', c 23.53; find a, b, 1', and K.
=
10. Given a = 96° 41.4', l' = 23° 13.3', a
= 2.458; find b, c, fJ and K.
__n__---------------------....---------
136
PLANE AND SPHERICAL
OBLIQUE
TRIGONOMETRY
11. Given {J = 40° 13/ 20", l' = 60° 12/ 13", b = 22.659; find a, c, and c..
12. Given {J = 18° 22' 26", l' = 99° 15/ 27", a = 35.863; find b, c, and a.
13. Given a = 68° 42/ 28", {J = 35° 42/ 18", a = 27.423; find b, c, and 1',
14. The distance between two points P and Q in a horizontal plane cannot
be measured directly.
In order to find the distance, a line P A = 238 ft.
is measured in the same plane, and the angles APQ = 128° 38/ and P AQ =
35° 58' are measured.
Find PQ.
Ans. 526.37 ft.
15. To find the width of a river, a line AB = 600 ft. is measured on
one side parallel to the bank of the stream.
A tree C stands on the opposite
bank.
The angles ABC = 65° 30', and BAC = 81° 10' are measured.
Find the width of the stream if line AB is 30 ft. from the bank of the stream.
Ans. 951.8 ft.
16. Find the area of a triangular
plot of ground one side of which is
130 rd., and the angles adjacent to this side are 47° 15' and 55° 45'.
Ans. 5264 sq. rods.
,and
.!L.~
5;,
~
(J)
~
a
'.
'--
a /
./
\z
\a
(1)(b)
b
b
S!.--
.
18. The points A and B are on opposite sides of a river, and the distance
AB cannot be measured directly. A point C is chosen on the same side of
the river as A and the following measurements made: AC = 600 ft., LCAB
10/. Compute the distance AB.
= 80° 45/, and LACB = 60°
Ans. 825.6 ft.
L-
c ,
mea)
c sin {3
b =
sin (a + (J)
137
(b) Angle is obtuse and opposite side not greater than
adjacent side.
(2) One solution when:
(a) Angle is acute and opposite side is equal to adjacent
side times the sine of the angle. This gives a right
triangle.
(b) Angle of any size and opposite side greater than adjacent side.
17. In a triangle, given c, a, and {J;prove that
c sin a
a =
sin (a +
TRIANGLES
c
f....--
!L-
~a
~
~L
b
(2)(a)
(2) (b)
b
a
f.
93. Case II. The solution of a triangle when two sides and
an angle opposite one of them are given.-It
is known from
geometry that when two sides and an angle opposite one of them
are given the triangle may net be uniquely determined.
With these parts given: (1) It may not be possible to construct
any triangles;
(2) it may be possible to construct
just one triangle; (3) it may be possible to construct
two triangles-the
ambiguous case.
EXERCISES
Construct carefully the following triangles:
1. (a) a = 1 in., c = 3 in., and a = 40°.
(b) a = 2 in., c = 3 in., and a = 140°.
2. (a) a = 1 in., c = 2 in., and a = 30°.
(b) a = 3 in., c = 2 in., and a = 35°.
(c) a = 3 in., c = 2 in., and a = 120°.
3. a = 2 in., c = 3 in., and a = 30°.
Corresponding to Exercises 1, 2 and 3 above, we have the
following, which should be compared with the corresponding constructions in Fig. 88
(1) No solution when:
(a) Angle is acute and opposite side less than adjacent
side times the sine of the angle.
~
(3)
Fw.8S.
(3) Two solutions when angle is acute and the opposite side
greater than the adjacent side times the sine of the angle, and less
than the adjacent side.
The ambiguity
d (3) is also apparent from the solution of l' found from
c sin a..
the relation sin 'Y =
This equation has two values of l' less than 180°
a
each of which may enter into the triangle when a is acute.
With each of
these values of l' there may be found values of {3and b, thus making two
triangles.
When logarithms are used, proper conclusions can be drawn from the following, where a, b, and a are given.
For other given parts, the proper change
can easily be made.
If log sin {3 = 0, sin {J = 1, {3 = 90°; hence a right triangle.
If log sin {J > 0, sin {J > 1, which is impossible; hence no solution.
If log sin {J < 0 and b < a, and therefore {3 < a, only the acute value of {J
can be used; hence there is one solution.
If log sin {3 < 0 and b > a, both acute value of {Jand its supplement may
be used; hence there are two solutions.
138
PLANE
AND
SPHERICAL
OBLIQUE
TRIGONOMETRY
If the given parts are a, c, and a, with a acute and a < c, the
formulas for the solution are:
~
sm
= ~,
a
sm 'Y
{3 = 1800
which gives two values
-
(a
b
a
.
b
c
.
- sIn a gIves
~
sIn I-' - ---;--
+ 1');
{3'
b,. ~ b'
=
1800
(a
.
+ 'Y');
- --;---a giVes, b'
SIn I-' - sIn
c
.
.
were
h
Example.-Solve
a = 340 15.3'.
Solution.-Here
are two solutions.
the triangle
when a
=
-
11.75, c = 15.61, and
= 11.75.
GiVen{~ = 15.61.
970 20.8'
Apply the
1. Given
2. Given
3. Given
4. Given
5. Given
6. Given
find
tests
a =
a =
b =
b =
a =
a =
log a =
log sin {3 =
colog sin a =
log b =
b =
log a =
log sin {3' =
colog sin a =
log b' =
b' =
1.07004
9.99642
0.24959
1.31605
20.704
1.07004
9.38801
0.24959
0.70764
5.1008
EXERCISES
and determine the number of solutions in Exercises 1 to 5.
4, e = 5 and a = 55°.
25, b = 45 and {3 = 117°.
72, e = 28 and 'Y = 21°.
22.5, e = 55.3, and {3 = 24° 0.5'.
49.7, b = 55.3, and a = 132°, 20.5'.
78.291, e = 111.98, a = 38° 21.3';
'Y= 62° 34.1', {3= 79° 4.6', b = 123.88.
'Y' = 117° 25.9', {3' = 24° 12.8', b' = 51.74.
7. Given a = 84.675, b = 94.423, {3 = 69° 11' 28";
find
'Y = 53° 51' 2", a = 56° 57' 30", e = 81.564.
8. Given a = 16.1, b = 18.7, and a = 22° 18' 23";
= 340 15.3'.
'Y = 48023.9'.
hnd
20.704.
131036.1'.
1408.6'.
5.1008.
find
Formulas
c sin a_.
,
a
c
. .'. SIn 'Y -- - a - sIn 'Y.
-,-- = -,-sm a
sm 'Y
{3 = 1800 - (a + 'Y); {3' = 1800 - (a + 'Y').
a sin {3.
~ =~
. .. b =
sin a
sin {3 sin a
sin {3'
. . . b'=-.a sin
- b' {3' =-sina a
a
sin
Logarithmic formulas
log sin l' = log c + log sin a + colog a = log sin 'Y'.
+ log
+ log
= 131036.1'
'Y
a is acute, a < c, and a > c sin a; hence there
log b = log a
log b' = log a
'Y'
({3 + 'Y),
Construction
=
'Y' =
{3' =
b' =
= 480 23.9'
{3' = 140 8.6'
The area K can be determined as follows: suppose b, c, and
Then K = tbh = tbc sin a, and a = 1800
.
b sin 'Y
.
R
I-' can b e d etermme d f rom sm {3 = -. c
l'
. . . {3 =
b . b' - ---, , gIves. b'
or ~
l'
sm 'Y gIves ,~,sm I-' - sm
sm I-' - --;--are given.
139
Computation
log c = 1.19340
log sin a = 9.75041
colog a = 8.92996
log sin l' = 9.87377
for 'Y, say, 'Y and 'Y';
-
a
TRIANGLES
sin {3 + co log sin a.
sin {3' + colog sin a.
{:J= ~(j" B' ~B", 'Y = 1,H" J~' t'\", c = Jl./b~.
{3' = 153° 50' 31", 'Y' = 3° 51' 6", e' = 2.849.
9. Given a = 58.345, b = 47.654, and {3 = 18° 15' 46";
a = 22° 33.7', 'Y = 139° 10.5', e = 99.415.
a' = 157° 26.3', 'Y' = 4° 17.9', e' = 11.4.
10. Given a = 248.4, b = 96.1, a = 66° 31';
find
e = 270.5, {3 = 20° 47', 'Y = 92° 42'.
11. Given a = 462.3, b = 535.9, a = 42° 32';
find
e = 682.07, {3= 51° 35.7', 'Y = 85° 52.3'.
e' = 107.7, {3' = 128° 24.3', 'Y' = 9° 3.7'.
12. Given b = 160, C = 180, {3= 20° 18' 23";
find
a = 316.1, a = 136° 42' 47", 'Y = 22° 58' 50".
a' = 21.51, a' = 2° 40' 27", 'Y' = 157° l' 10".
13. Given b = 32.597, e = 43.465, and {3 = 28° 43.6';
find
a = 63.14, a = 111 ° 25', 'Y = 39° 51.4'.
a' = 13.092, a' = 11° 7.8', 'Y' = 140° 8.6'.
14. Given b = 46.342, e = 65.899, and {3 = 21 ° 15' 18"
find
a = 101.13, a = 127° 42' 50", 'Y = 31
° l' 52".
a' = 21.706, a' = 9° 46' 34", -y' = 148° 58' 8".
15. Given a = 24.897, b = 33.543, a = 26° 44.9'; find e, {3,and 'Y. Check
I:y \1o11weide's eauati0lli!
J
,
16. Given a = 25.34, c = 45.76, a = 35° 43.8'; find b, 13,and -y. Check.
17. Given b = 366.62, C = 621.35,13 = 154° 38'; find a, a, and -y. Check.
18. Given a = 322.22, C = 847.36, a = 17° 34' 48"; find b, 13,and -y.
Check.
By
94. Case III.
The solution of a triangle when two sides and
the included
angle are given.
First method.-Let
the given
parts be a, b, and 'Y. Then, from the law of cosines,
[26] and
'Y d
= a- sin
an
c
As a check, a
+ +
{3
'Y
.
sIn
(.I
IJ
b sin
= -, c
'Y
Substituting
.
(3)
+ (3 = 1800 -
this in [37] gives
1
Formula
[37] or [38] makes
a-b
a
+
b
it possible
1
cot 2Y'
to find
tea - (3) when
Ha
+ (3) = 900 - h, therefore
a and (3 can be found
from the
relatio~s :
a = Ha
+ (3) + !(a - (3),
(3 = !Ca + (3) - Ha - (3).
It is evident that the other side can be found by the law of sines,
which may also be used as a check, together with a + (3+ 'Y = 1800
flftpr finding !'Y. A more certain check formula is one of Mollweide's equations.
A discussion similar to the above can be given when any two
sides and the included angle are given. The other formulas can
also readily be written by a cyclic change in the letters.
A convenient set of formulas for solving the triangle when a,
b, and 'Yare given is
Ha + (3) = 900 - h.
and
-.
96. Case III. Second method.-For
a solution by logarithms
when two sides and the included angle are given, the following
theorem, known as the law of tangents, is needed.
LAW OF TANGENTs.-In any triangle the difference of any two
sides is to their sum as the tangent of half the difference of the opposite
angles is to the tangent of half their sum.
a
sin a
.
Proof.- = ~ {3 from the law of smes.
sm '
b
a - b
sin a - sin {3
.
Then =. SIn a +. sIn (3' by a theorem of proportIOn.
a + b
(3)
a, b, and 'Yare given, while Ha + (3)can readily be found because
K = tab sin y.
following by the first method and check:
a = 2, b = 3, -y = 41 ° 39.8'; find c, a, 13,and K.
a = 4, c = 8, 13= 105° 32.3'; find b, a, -y, and K.
b = 27, C = 80, a = 64° 45' 34"; find a, 13,-y, and K.
a = 19, b = 29, -y = 76° 24'; find c, a, 13,and K.
b = 14, C = 16, (t = 125° 18.9'; find a, 13,'Y,and K.
-
form for
tan 2(0: - ~) =
[38]
= 1800 may be used, or use Mollweide's
It is evident that the formula for finding c is not adapted to the
use of logarithms.
This method is often convenient, however,
when the numbers expressing the sides contain few figures or when
only the third side is to be found.
Solve the
1. Given
2. Given
3. Given
4. Given
6. Given
-
(3) = 900 - h,
'Y and tea +
(3)
.'. tan tea + = tan (900 - h) = cot h.
.
respec t lve 1y.
EXERCISES
!Ca + (3)sin tea
141
tan !Ca - (3)
= tan tea - (3)cot tea + (3) = tan tea + (3)'
.'. a - b = tan!( 0: - ~).
a + b
tan t(o: + ~)
a
equations.
The area K = thb = tab sin 'Y; or, in words, the area equals
one-half the product of the two sides and the sine of the included angle.
[36]
2 cos
= 2 sin tea + (3)cos tea
This can be put in another
and a and {3may be found from
.
[26],
[37]
c = vi a2 +b2 - 2ab cos 'Y,
SIn a
TRIANGLES
OBLIQUE
PLANE AND SPHERICAL TRIGONOMETRY
140
!
1
1
(3) = -a - b cot -1'
tan -(a
2
a+b
2'
a = !Ca + (3)+ !Ca - (3).
-
(3 = !Ca + (3) - !Ca - (3).
b sin 'Y
a sin
c = - sin al' =-' sin {3
It should be noted that negatives are avoided if the larger angle
and side come first in [38]. Thus, if {3 > a and hence b > a,
write [38] in form tan ~({3- a) = ~ ~ ~ cot ~I"
~
142
PLANE
AND
SPHERICAL
Example.-Solve
the triangle
and f3 = 79° 31' 44".
Solution.
a
Given c
{ f3
OBLIQUE
TRIGONOMETRY
when a = 42.367, c = 58.964,
Construction
=
=
42.367.
58.964.
= 79° 31' 44".
B
~
c
b = 66.057.
To find a = 39° 6' 1".
{ l' = 61° 22' 15".
A
a
ex
l' C
b
Formulas
101
1
c
-
a
1
,i'Y + a) = 90 - 2f3,tan 2(1' - a) = c + a cot 2f3,
asin f3
c- a
sin Hl' b
d f
h k,
= sm
;---'
a' an
or a c ec
- b
=
cos tf3
a) .
Computation
u.
=
=
=
=
ItJg_0
58.964
42.367
16.597
101.331
tf3 = 39°45'52"
nuL==D200-3
colog (c + a) = 7.99426
log cot tf3 = 0.07981
log tan H'Y
H'Y
H'Y
log a = 1.62703
log sin f3 = 9.99270
colog sin a = 0.20019
log b = 1.81992
b = 66.057
Check
-
a)
=
-
_nnn
c
a
c- a
c+a
9.29410
- a) = 11° 8' 7"
+ a)
= 50° 14' 8"
l'
61° 22' 15"
=
u.
..
togQ.
== ..
log sin H'Y - a) =
colog cos tf3 =
log (c
6. Given b = 248.65,
c = 471.69, and ex = 139° 8' 46";
a = 679.52, fJ = 13° 50' 55",
l' = 27° 0' 19", K = 38,361.
7. Given a = 43.5, b = 38.1, l' = 57° 14.9'; find c, ex, and fJ. Check
by Mollweide's equations.
8. Given a = 26, c = 25, fJ = 42° 56.8'; find b, ex, 1', and K. Check.
9. Given b = 569.59, c = 543.76, ex = 71 56';
°
find
a = 654.21, fJ = 55° 51.9',
l' = 52° 12.1'.
10. In an isosceles triangle each of the two equal sides is 23 in. and the
included angle is 58° 40'. Find the third side.
Ans. 22.5 in.
11. The two diagonals of a parallelogram
are, respectively,
30 and 25 in.,
and one of the angles formed by them is 71
° 25'. Find the sides of the
parallelogram.
Ans. 16.18 in.; 22.38 in.
12. To find the distance AB through a swamp, a point C was chosen and
the following measurements
made: CA = 163 rd., CB
= 145 rd., and angle
ACB = 36° 37'. Compute the distance AB.
Ans. 98.25 rd.
13. At a certain point the length of a lake subtends an angle of 53° 44.5',
and the distances from this point to the extremities of the lake are 144 and
86.3 rd., respectively.
Find the length of the lake.
Ans. 116.1 rd.
14. Two railroad tracks intersect at an angle of 85° 30'. At a certain
time a train going 32 miles an hour passes the point of intersection;
2 min.
later a train going 55 miles. an hour on the other track passes this point.
Write a formula showing their distance apart t min. after the first train
passes the intersecting
point.
How far will they be apart in 25 min.?
Ans. 24.045 miles, or 25.815 miles.
16. Two headlands P and Q are separated by water.
In order to find the
distance between them a third point A is chosen from which both P and Q
are visible,
and the following
measurements
are made:
AP
= 1160 ft.,
A Q = 1945 ft., and angle P A Q = 60° 30'. Find the distance PQ.
Ans. !7T)5 ft
----..-------
96. Case IV. The solution of a triangle when the three sides
are given.-In
this case the angles can be found by means of
the law of cosines, from which the following formulas are derived:
- a) = 1.22003
cos a =
a = 39° 6' I"
cos f3
=
cos
=
EXERCISES
1. Derive formulas like [381 for finding tan Hex
1'); for tan H1' 2. Givena = 50.35, b = 36.54, l' = 125° 12.3';
find
c = 77.405, ex = 32° 6.4', fJ = 22° 41.3'.
3. Given a = 26.548, c = 41.654, fJ = 61
° 0' 33";
find
b = 36.986, ex = 38° 53' 29",
l' = 80° 5' 57".
4. Given a = 51.455, b = 27.345, and
l' = 51 ° 19.8';
find
c = 40.461, ex = 96° 49.3', fJ = 31 50.9', K = 549.29.
°
6. Given a = 285.6; b = 171.4, and
l' = 65° 41' 10";
find
c = 265.78, ex = 78° 19' 9", fJ = 35° 59' 41", K = 22,305.
143
find
1.8J~2.
9.28585
0.11426
c - a = 16.597
-
TRIANGLES
fJ).
l'
b2
+ c2 -
a2
2bc
+ c2 2ac
a2 + b2 -
a2
2ab
b2
.
c2
.
These formulas give the cosines of the angles and, therefore,
the angles; but they are not adapted to logarithms.
They are
convenient when t.he sides are expressed in numbers of few
figures, or when tables of squares and products are at hand
A very good check formula is a + f3+
l'
= 180°.
l
PLANE
144
AND
SPHERICAL
OBLIQUE
TRIGONOMETRY
EXERCISES
[411]
Find
1. a
2. a
3. a
4. a
the angles when the sides are given as follows:
5. a = 10, b = 8, and c = 7.
= 3, b = 4, and c = 5.
3,
and
c
4,
b
=
6.
6. a = 200, b = 300, and c = 400.
=
=
19,
and
c
21.
7.
a = 12, b = 17, and c = 14.
15,
b
=
=
=
13,
and
c
16.
8.
a = 12, b = 5, and c = 13.
12.
b
=
=
=
97. Case IV. Formulas
adapted to the use of logarithms.e2 - a2
b2
(1) Start with the equation
each member of it from 1.
1
-
cos a
. 21
.,. 2 sm"2a
= 1-
a2 -
b2
+
+
=
cos a
[412]
[413]
-
e2
(a
-
+
b
e
+ e) (a + b 2b
2(s
=
-
[391]
!n-like
.'. Slll 2a =
manner
2(s
-
b)2(s
-
e)
by writing
b), and
!lrR ohtHinRrl thR followinp:'
[393]
= ~(s
sin
!r
2
-
In using any ()!~-~~~~~s off()~:r!l~las, the work ~ay ~{Jc~e_(}./({J<!__
OJ!
-------------
(2) By adding each mem ber of the equation cos a =
!a
1
=
S(s
[402]
cos -1ftIf = ~S(s
2
1
~
-
a)
be
-
b2
+ e2
-
a2
.
.
S(s - e)
.
ab
ae
b)
Cos -r =
2
(3) By dividing each formula of the set under (1) by the corresponding formula of the set under (2), there results:
[403]
~
+ !/3 + h
= 90°.
The area can be found from
2be
to 1, and carrying out the work in a manner similar to the above,
there are obtained the following:
2
b)(s - c),
1
~.
tan 2"~=
s - b
1
tan2"r=s-e~.
[423]
ae'
- a)(s - b)
ab'
COS-a
~
1
r
tan -a = -.
2
s - a
[421]
[422]
a)(s - e)
-
[401]
- a)(s - b)(s - e),
s
Similarly, the following are obtained:
V
[392]
e)
- a)(s - b)(s - c)
= ~(s
s(s - a)2
a)(s
r = ~(s -
e).
2bc
I(s - b)(s - e) .
-~i~ ~~ = ~(S
2
-
s(s
= s-~~(s
- a
Substituting these values in the above,
2 =
. 1
2r =
a2
Then a - b
2 sin2 !a
1
e).
tan 2~ -- ~(S - a)(s
s(s - b)
1
b).
~(S - a)(s
tan
s(s - a)
.
(b2 - 2be + e2)
-2be
Let a + b + e = 2s.
a + b - e = 2(s - e).
1
- f).
tan 2a --"\I"res -s(sq)(s
- a)
.
- b)(s - e)
.
Since ~(s
This gives
2be
145
These last three can be put in a form slightly more convenient
and subtract
2be
TRIANGLES
K = !bh = !be sin a = be sin !a cos !a
r7 (
~(s - b)(s - e) ~S(s - a) - V
- s~s
- a.)(
= be
be
be
[43]
.'.K = vs(s - a)(s - b)(s - e).*
Since the sine varies most rapidly for small angles, and the
cosine most rapidly for angles near 90°, formulas [39] should be
used when the angles are small, and [40] when the angles are
near 90°. In all cases the tangent varies more rapidly than either
sine or cosine. Hence, formulas [41] or [42] are always more
nearly accurate than [39] or [40].
* Formula [43] was discovered by Hero (or Heron)
the beginning of the Christian era.
of Alexandria
about
l
146
PLANE AND SPHERICAL TRIGONOMETRY
OBLIQUE
Again, formulas [41] or [42] are more convenient, since, for a
complete solution of the triangle, they require only four logarithms to be taken from the table; while [39] and [40] require,
respectively, six and seven.
B
c
A
Formulas [42] may be derived by taking from geometry the
fact that the area of a triangle, when the three sides are given, is
K = ys(s
-
a)(s
-
b)(s
-
Also
But
+
EC
+
Remark.-The
sum of 8, (8 - a), (8 - b), and (8 - c) is 2s,
and hence is a check on the additions and subtractions.
To facilitate the subtractions, write the values of 8 on the margin of a slip of paper, when it can be placed above the values a,
b, and c, successively.
In like manner log r can be written on a
margin and placed above logs of (s- a), (8 - b), and (8 - c).
.
. 1
1. D enve sm 2'Y
=
2. Derive cos ~tJ =
.
~ I(s - a) (s - b)
f rom t h e 1aw 0 f cosmes.
ab
"~~
from the law of cosines.
tqngpnt
,,~hpn
t.h~
~nr:ip
iq npf-IT
11:-"?
Hoi.\:"
RPPllrRt,p1y-
p:ln
t.hp
:lnf"1p
hf'
fOl1nrl
from each?
5. Answer the same questions for 82° and 46°.
6. In Fig. 89, show that BE = s - b.
7. Can s - a be less than O. Show why.
.
. 1
~ I(s - b)(s - c) ?
.
.
S. H ow many va 1ues 0 f a WI11sabs f y sm 2a =
EE = s.
9. In
It should be noted that r in the formulas of this article is the
radius of the inscribed circle, and the formula given for r is a
simple formula for finding the radius of the inscribed circle.
for the angles when a = 23.764, b
1.74936
0.87468
9.47868
16° 45' 21"
0.07693
50° 2' 53"
9.63196
23° 11' 45"
3. Derive K = vs(s - a)(s - b)(s - c) by geometry.
4. What is the tabular difference for each of log sine, log cosine, and log
. . . AF = s - (EC + EE) = s - a.
1
r
tan -a = -.
2
AF
1
r
.', tan -a =-.
2
s- a
Example.-Solve
and c = 31.166.
Solution.-Use
a =
b =
c =
28 =
log r2 =
log r =
log tan !a =
, . . !a =
log tan !(3 =
A check.
. . , !(3 =
log tan h =
... h =
Check.-1a + !(3 + h = 89° 59' 59".
EXERCISES
.'. sr = ys(s - a)(s - b)(s - c),
/(s - a)(s - b)(s - c) .
r =\1
Ii
AF
147
8 = 48.653
s - a = 24.889
8 - b = 6.277
8 - c = 17.487
28 = 97.306
c);
and, from Fig. 89, K = sr, where r is the radius of the inscribed
circle.
and
TRIANGLES
=
42.376,
be
them are given, an ambiguity was introduced because from the sine of the
angle two values of the angle were found.
Why is there not an ambiguity
when formulas [39] are used?
10. Given a = 72.392, b = 55.678, c = 42.364;
find
!a = 47° 6' 10", !i3 = 25° 2' 42", !'Y = 17° 51' 11".
11. Given a = 43.294, b = 40.526, c = 39.945;
find
formulas [42] with that for r.
23.764
log (s - a) = 1.39600
42.376
log (s - b) = 0.79775
31.166
log (s - c) = 1.24272
97.306
colog 8 = 8.31289
"
solving the triangle, when two sides and an angle opposite one of
!a = 32° 32' 45", !i3 = 29° 3' 2", !'Y = 28° 24' 11".
12. Given a = 610, b = 363, C = 493;
find
a = 89° 33' 50", tJ = 36° 31' 2",
'Y
= 53° 55' 6".
13. Given a = 16.47, b = 25.49, C = 33.77;
find
a = 28° 5' 2", tJ = 46° 46' 4", 'Y = 105° 8' 51".
14. Solve the example in Art. 97 by using formulas [39].
formulas
[40].
Compare
the work with that in the solution
By using
of the example.
l
148
PLANE
AND
SPHERICAL
OBLIQCE
TRIGONOMETRY
.
15. Given a = 98.34, b = 353.26, C = 276.49;
a = 11° 16' 58", fJ = 135° 20' 27", -y = 33° 22'32", K = 9,554.5.
16. Given a = 8.363, b = 5.473, C = 10.373;
find
a = 53° 27' 12", fJ = 31° 43' 8", -y = 94° 49' 40", K = 22.804.
17. Given a = 49.63, b = 39.65, C = 67.54; find a, fJ, -y,and K. Check.
18. Given a = 2.374, b = 4.375, C = 5.73; find a, fJ, and -yo Check.
GENERAL
K
149
= b2 sin a sin -y
2 sin (a
+
-y)'
16. Use the corollary of Art. 90, and the formula K = !ab sin -y, and show
that the radius of the circumscribed circle is given by R = abc Also show
4K'
K = abc.
4R
17. In a parallelogram
given a diagonal d = 15.36, and the angles a
=
26° 36.4', and fJ = 36° 32.4' which this diagonal makes with the sides; find
the sides.
Ans. 10.25, 7.711.
18. In a parallelogram
are given a side a, a diagonal d, and the angle (J
between the diagonals; find the other diagonal and side.
19. If one side of a parallelogram
is 13.52 in., one diagonal is 19.23 in.,
and one angle between the diagonals is 35° 32' 35", find the other diagonal.
Ans. 40.27 or 8.974 in.
20. The two parallel sides of a trapezoid are a and b, and the angles formed
by the nonparallel
sides at the two ends of one of the parallel sides are,
respectively, a and fl. Find the lengths of the nonparallel sides.
b) sin a
(a - b) sin fJ.
Ans. (a. and
sm (a + fJ)
sm (a + fJ)
21. The two parallel sides of a trapezoid are, respectively,
17.5 and 9.3 ft.,
and the angles formed by the nonparallel sides at the ends of the first side
are respectively
31 ° 25', and 52° 36'. Find the langths of the nonparallel
sides.
Ans. 4.298 ft., 6.55 ft.
22. Show that the area of any quadrilateral
is equal to one-half the
product of its diagonals and the sine of the included angle.
23. One side of a parallelogram is 46.4 rd., and the angles which the
diagonals make with that side are 57° 34' and 36° 34'. Find the length of
the other sidf'.
An~. 49.67 rd.
24. Two circles whose radii are 28 and 36 in. intersect.
The angle
between the tangents at a point of intersection is 36° 35'. Find the distance
between their centers.
Ans. 60.82 in., 21.47 in.
25. B is 48 miles from A in the direction N 71 W, and Cis 75 miles from
°
A in the direction N 15° E. What is the position of C relative to B?
Ans. 86.18 miles, N 48° 45' 16" E.
26. Given a parallelogram
ABCD with AD
= m, AC = d, AB = n,
(q,
m
- a)
LBAD = q" and LDAC
a; prove that. sin
and that
that
19. Given a = 70, b = 40, C = 35; find the area of the triangle and the
radius of the inscribed circle.
Ans. K = 470, r = 6.483.
20. The sides of a triangle are, respectively
28, 16, and 25 ft. Find the
area of the triangle and the area of the inscribed circle.
Ans. 198.52 sq. ft., 104.02 sq. ft.
21. Find the radius of the largest circular gas tank that can be constructed
on a triangular lot whose sides are 75, 85, and 95 ft., respectively, and locate
the center by giving the distance from the ends of the 85-ft. side to the
point of tangency on the other sides. Ans. r = 23.85 ft., 52.5 ft., 32.5 ft.
EXERCISES
1. Find the area of a triangle with sides 13.6 and 16.39 ft. and included
angle 163° 36' 16".
Ans. 31.459 sq. ft.
2. Find the area of a triangle with the three sides, respectively,
47.45,
36.4, and 36.65 ft.
Ans. 658.85 sq. ft.
3. Two sides of a parallelogram
are 46.3 and 46.36 rd., respectively,
and the included angle is 56° 35'. 'Find the area. Ans. 1791.6 sq. rd.
4. The base of a triangle is 62.53 ft. and the two angles at the base are,
respectively,
109° 53', and 36° 16'; find the other two sides and the area of
the triangle.
Ans. 66.407 ft., 105.57 ft., 1952.5 sq. ft.
5. Two angles of a triangle are, respectively, 57° 47' 14" and 59° 47' 43".
If the included side is 14.63 in., find the area.
Ans. 88.286 sq. in.
6. Tn a trianglf' an angle is 52° 16' and thp oppositf' side is 36 in,; find thp
diameter of the circumscribed
circle.
Ans. 45.52 in.
7. If the sides of triangle are 4, 6, 7, find the radius of the inscribed
circle.
Ans. 1.41.
8. If the sides of a triangle are 4, 6, 5, find the radius of the circumscribed
circle.
Ans. 3.024.
9. The three sides of a triangle are 8, 12, 15; find the length of median
drawn to the side 12.
Ans. 10.42.
10. In a triangle ABC, angle A is 126° 47', and AD is the bisector of angle
A with D on the side BC.
If b = 24, and C = 15, find AD, BD, and DC,
Ans. AD = 8.27, BD = 13.5, DC = 21.6.
11. The angles of a triangle are in the ratio of 3: 5: 7; and the longest side
is 154 ft. Solve the triangle.
Ans. Angles, 36°, 60°, 84°; sides 91.02, 134.1.
12. The sides of a triangle are in the ratio of 7: 4: 8; find the sine of the
smallest angle.
The cosine of the largest angle.
Ans. 0.49992, 0.01786.
13. Solve the following triangle for the parts not given: K = 7934.2,
a = 36°, and fJ - -y = 16°.
Ans. a = 102.65, b = 171.99, C = 156.97, fJ = 80°, -y = 64°.
14. The sides of a triangular field of which the area is 13 acres are in the
ratio of 3;4;6.
Find the length of the shortest side.
Ans. 59.248 rds.
.
15. Prove that ill any tnangle
find
rUUNGLES
..
_!
=
cot q, = cot a
sm q,
-
m
d sin
a'
= -d
If AD = AB =
m,
prove that d = 2m cos !q,.
27. Given two triangles with data shown in Fig. 90; prove that p
= w
tan 50°. If w = 200, find the values of p, rl, r2, and r,.
Ans. p = 238.36, rl = 178.46, r2
= 300.56, r, = 254.87.
EXERCISES, APPLICATIONS
1. Two streets intersect at an angle of 86° 36'. The corner lot fronts
100 ft. on one street and 146 ft. on the other, and the other two sides are
perpendicular to the streets.
Find the area of the lot.
Ans. 13,696 sq. ft.
~
150
PLANE
AND
SPHERICAL
OBLIQUE
TRIGONOMETRY
p
.....
p
/"
/"
D
A
x
-/""
.-f.;q,
~
?-a
a
y
B
FIG. 91.
C
I
""
I
/
Ix
=
asin(3tanq,
asinatanO
sin (a + (3) = sin (a + (3)
/
,"
/
/
/90°'-
~
I
/.
t--i,t
I 90}"7C
I
131..8 /"/"
---j)-'Y
'a
" <,: a
'Y' "
B
FIG. 92.
-7
/
I
190of
I /'
If) /
:<
I
I
, <...
"'I"
A
"II
I
/"
/'
I
6. In Fig. 92, the point P is an inaccessible object above the horizontal
plane ABC.
The straight line AB = a is measured, also the angles a, (3, 0,
and q,. Find the height x of the point P above the plane, giving two solutions which will check each other.
State the result in the form
x
151
8. Two observers at A and B, 125 rd. apart on a horizontal plane, observe
at the same instant an aviator.
His angle of elevation at A is 72° 25', and
at B 64° 34.8'.
The angles made by the projections of the lines of sight on
that horizontal plane with the line AB are 40° 27' at A and 25° 38' at B.
Find the height of the aviator.
Ans. 3080 ft.
9. Compute the inaccessible distance PQ (Fig. 93) when given the line
AB = a and the angles a, (3, 'Y, and o. Are
the data sufficient for a check?
10. In Exercise 9, given a = 330 ft., a =
41 ° 36.5', (3 = 64° 47.5', 'Y = 30° 46.5', and
0 = 35° 53.5'; find PQ.
Ans. 271.8 ft.
11. To find the distance
between two
inaccessible points A and B, a base line CD
= 800 ft. is measured in the same plane as
A and B, and the angles DCA = 106°, DCB
= 39°, CDB = 122° and CDA = 41° are
measured. Compute the distance AB.
FIG. 93.
Ans. 1924 ft.
12. Two points P and Q are on opposite sides of a stream and invisible
from each other on account of an island in the stream.
A straight line AB
is run through Q and the following measurements
taken: AQ = 824 ft.,
QB = 662 ft., and QAP = 42° 34.4', and angle QBP
= 57° 45'. Compute
QP.
Ans. 872.1 ft.
2. Along a bank of a river, a line 500 ft. in length is measured.
The
angles between this line and the lines drawn from its extremities to a point
P on the opposite bank of the river are, respectively,
62° 35' and 55° 44'.
Find the width of the river.
Ans. 416.7 ft.
3. A bridge is to be constructed over a valley.
If the length of the bridge
is 1 and the inclinations
of the two sides of the
valley are respectively
a and (3, find the height of
a pier erected at the lowest point of the valley to
.
lsin a sin (3.
support the bndge.
Ans.
sin (a + (3)
4. A ship at a point Q observes two capes A
and Bj the bearing of A is N 36° 35' E, and the
bearing of B is N 16° 36' W. Find the distance the
ship is from each cape if it is known that the
distance between the capes is 23.8 miles, and the bearing of B from A is N
58° 40' W.
Ans. 19.92 miles from A; 29.61 miles from
y ofB. an
6. In Fig. 91, find the height DC = x, and the distance AC =
inaccessible object, having measured on a horizontal plane the distance a
in the line CAB, and the angles a and (3.
~sin(3
AD =
Suggestion.sin (a - (3)'
a sin a sin (3
CD = AD sin a = sin (a
- (3)'
a cos a sin (3.
AC = AD cos a =
sin (a - (3)
,/'
TRIANGLES
1//
//
{3 /
"B '('.\-..,,/
,\ /
V
D
FIG. 94.
lE
I'
LLHC
"
9O~V
/
A
FIG. 95.
13. Two points P and Q on the same side of a river are inaccessible.
They
are both visible from a single point A only, on the opposite side of the river.
From other points on this side of the river only P or only Q can be seen.
Show what measurements
can be made to compute PQ, and outline the
solution.
14. A statue, of height 2h, standing on the top of a pillar, subtends an
angle a at a point on the ground distant d from the foot of the pillar.
Prove
that the height of the pillar is yh2 + 2hd cot a - d2 - h.
.
(3 = 52° 46' 30",
7. In Exercise 6, given a = 465 ft., a = 49° 51' 47",
t/>= 39° 16' 14", and 0 = 40° 25' 5"; find x. Could this exercise be solved if
Ans. 310.26 ft.
()
were not given?
-.
~
152
PLANE
AND
SPHERICAL
TRIGONOMETRY
15. A flagstaff 50 ft. tall stands on the top of the end of a building 105 ft.
high.
At what distance from the base of the building will the flagstaff
subtend an angle of 9°?
Ans. 251 ft. or 64.9 ft.
16. In taking measurements
for finding the height of P (Fig. 94) above
the horizontal line AC, a line AB = a was measured in a plane making an
angle DAB = "I with the horizontal.
Other angles measured were LDAC =
a, LADC = ~, LCAP = t/>,and LEBP =
above C, and put in the form
o.
Find the height
x that
Ans. 899 ft.
-.D
......
P is
......
"""
,/
~
,I
f
.LJ..
q) !
B
""',
"""',
b
'........
"
'IA
A
28. Given the data as shown in Fig. 96; find the distance
(a
+ x)(b + x)
sin a sin ~
= ab sin
c
m
FIG. 97.
FIG. 96.
(a
+ "I)
x in form:
sin (~ + "I).
After numerical values are substituted,
this can be solved as a quadratic
equation in x.
29. Given data as shown in Fig. 97; solve for x and state the result in a
formula.
Ans. x = m[tan (a + ~)
- tan {1].
.
...
E
!
E
D
FIG. 99.
30. Given data as shown in Fig. 98; solve for x and state the result in a
(b sec a - m) sin ~
formula.
Ans. x
.
=
cos (a +) {1
31. A flagstaff of known height c stands on the top at the end of a building.
At a point P on the level with the base of the building, the building and the
flagstaff each sub tend an angle a. Find the distance of the point P from the
.
27. A kite K, sent up and fastened to the ground at a point A, drifted so
that it stands directly over the point B in the same horizontal plane as A
base of the building.
A ns.
tan
I
l
a
"
P~'7tJl.!
'"
......
"""- ---
/'
~~..y-
17. In Exercise 16, given a = 145 ft., a = 47° 60' 33", ~ = 60° 44' 20",
t/>= 59° 35' 12", "I = 4° 15' 31", and 0 = 63° 45' 43"; find x.
Ans. 226.94 ft.
18. From the data given in Fig. 95; find x and y in the forms:
asinacos{3
asinasin~
,y
.
x =
= sin (~
sin (~ - a - Ii)
- a - Ii)
19. From the top of a hill 720 ft. high, the angles of depression of the top
and the base of a tower are, respectively,
38° 30' and 51 ° 25'. Find the
height of the tower.
Ans. 263.1 ft.
20. A tower 120 ft. high casts a shadow 148 ft. long upon a plane which
slopes downward from the base of the tower at the rate of 1 ft. in 12 ft.
What is the angle of elevation of the sun?
Ans. 41 ° 53.5'.
21. A flagstaff 40 ft. high stands on the top of a wall 29 ft. high.
At a
point P on the level with the base of the wall and on a line perpendicular
to
the wall below the flagstaff, the height of the wall and the flagstaff subtend
equal ang;les. Find the distance of P from the wall.
Ans. 72.63 ft.
22. A tower stands on the top of a hill whose side has a uniform inclination
of 0 with thc horizontal.
At a distance of d from the foot of the tower
measured down the hill the tower subtends an angle t/>. Find the height h
d sin t/>
of the tower.
Ans. h =
cos (t/> + of
23. In the preceding exercise, find the height of the tower if 0 = 18° 45',
Ans. 224 ft.
t/>= 23° 45', and d = 410 ft.
24. From a point 250 ft. above the level of a lake and to one side, an
observer finds the angles of depression of the two ends of the lake to be
4° 15' and 3° 30', respectively.
The angle between the two lines of sight
is 48° 20'. Find the length of the lake.
Ans. 3128 ft.
25. A man is on a bluff 300 ft. above the surface of a lake.
From his
position the angles of depression of the two ends of the lake are 10° 30' and
6° 45', respectively.
The angle between the two lines of sight is 98° 40'.
Find the length of the lake.
Ans. 3239 ft.
26. From a point h ft. above the surface of a lake the angle of elevation
of a cloud is observed to be a, and the angle of depression of its reflection
in the lake is~.
Find that the height of the cloud above the surface of the
.
sin (~ + a)
I a k e IS h
a) ft
sin (~
153
and separated from it by water so that AB cannot be measured directly.
To find the height of the kite, a line AC 1000 ft. long is laid off on the horizontal, and the angles BAK = 46° 35' 52", KAC
= 67° 54' 39", and
ACK = 65° are measured.
Compute the vertical height of the kite.
sin ~ tan t/> a co~ "I sin a tan 0
x = ~~os. "I
+ a sin "I.
=
sm (a + m
sm (a + ~)
-
TRIANGLES
OBLIQUE
~
an ex
2 a Ct'
~
154
PLANE
AND
SPHERICAL
OBLIQUE
TRIGONOMETRY
a sin a sin {3
+ (3) sin (a - (3)'
(a
33. Two railway tracks intersect making an angle of 70°. The tracks are
connected by a circular Y that is tangent to each of the tracks at points
700 ft. from the intersection.
Find the radius of the Y and its length.
Neglect the width of the tracks.
Ans. 490.16 ft., 941.02 ft.
34. In Fig. 99, CE is parallel to DA, DA = 10 ft. AB = 10 ft., BC =
5 ft. and angle DAB = 120°. Find AE and angle AEC.
Ans. 18.03 ft., 46° 6.2'.
35. Two forces of 75 and 92 lb., respectively, are acting on a body.
What
is the resultant force if the angle between the forces is 54° 36'?
Ans. 148.6 lb.
36. Resolve a force of 250 lb. acting along the positive x-axis into two
components
of 170 and 180 lb., and find the directions of the components
with respect to the x-axis.
Ans. 46° 2' 23"; -42° 50'.
37. Two forces of 35 lb. each are acting on a body.
One is directed downward and the other at a positive angle of 47° with the horizontal.
Find the
magnitude of the resultant and its direction with reference to the horizontaL
Ans. 25.655 lb.; -21° 30'.
ance of the air, where will the bomb strike
the ground?
Ans. 1551.7 ft. east of point where bomb was
dropped.
Suggestion.-To
find the number of
seconds it is falling, use the equation
38. Three forces of 18, 22, and 27 lb. respectively,
and in the same plane
are in equilibrium.
Find the angles they make with each other.
Check
by noting the sum of the angles is 360°.
39. Four forces are acting on the origin of a system of rectangular
axes.
One of 300 lb. af'tR aclong the negactive x-axis. one of 175 lb. acts along the
positive x-axis, onecof 60 lb. acts at an angle of 50° with the x-axis, and one
of T lb. acts at an angle 8 with the x-axis.
If the forces are in equilibrium,
find T and 8.
Ans. -28° 0' 10"; 97.9 lb.
40. Five forces in equilibrium
are acting at the origin of a system of
rectangular
axes.
One of 4000 lb. acts along the negative y-axis, one of
1700 lb. acts along the negative x-axis, one of 1400 lb. acts at an angle
of 135° with the x-axis, one of T, lb. acts at an angle of 60° with the x-axis,
and one of T2 lb. acts along the positive x-axis.
Find T, and T2.
Ans. 3475.8 lb.; 952.1 lb.
41. An automobile is traveling N. 45° W. at 40 miles per hour, and the
wind is blowing from the northeast
at 30 miles per hour.
What velocity
and direction does the wind appear to have to the chauffeur?
Ans. 50 miles per hour N 8° 7.8' W.
42. A train is running at the rate of 40 miles per hour in the direction
S. 55° W., and the engine leaves a steam track in the direction N. 80° E.
The wind is known to be blowing from the northeast;
find its velocity.
Ans. 29.47 miles per hour.
43.
wind
mine
effect
In a river flowing due south at 3 miles per hour a boat is drifted by a
blowing from the southwest at the rate of 15 miles per hour.
Deterthe position of the boat after 60 minutes if resistance reduces the
of the wind 60 per cent.
Ans. 4.42 miles N, 73° 40.8' E.
155
44. A ship S is 12 miles to the north of a ship Q. S sails 10 miles per hour
and Q 15 miles per hour.
Find the distance and direction Q should sail
in order to intercept S which is sailing in a northeasterly
direction.
Ans. 29.23 miles, N 28° 7.5' E.
45. A tug that can steam 13 miles per hour is at a point P. It wishes to
intercept a steamer as soon as possible that is due east at a point Q and
making 21 miles per hour in a direction S. 58° W. Find the direction the
tug must steam and the time it will take if Q is 3 miles from P.
Ans. S 31 ° 7' 44" E; 7' 20.3".
46. Two poles are 42 ft. apart and one is 6 ft. taller than the other.
A
cable 48 ft. long is fastened to the tops of the poles and supports a weight of
400 lb. hanging from it by a trolley.
When the trolley is at rest find the
two segments of the cable and the angle each makes with the horizontal.
Suppose the"-trolley has no friction and that the two segments of the cable
are straight lines.
Ans. 30.2 ft., 17.8 ft.; each angle = 28° 57.3'.
Suggestion.-Tension
in cable is same throughout,
and horizontal components are equal.
47. An airplane, which is at an altitude of 1800 ft. and moving at the rate
of 100 miles per hour in a direction due
C
east, drops a bomb.
Disregarding the resist-
32. At each end of a horizontal base line of length 2a, the angle of elevation
of a mountain peak is {3,and at the middle of the base line it is a. Show
that the height of the peak above the plane of the base line is
vsin
TRIANGLES
~gt2
=
1600.
D
48. In a dredge del'l'ick (Fig. 100) the
FIG. 100.
following measurements
are made: AF is
perpendicular
to DE, AB = 20 ft., BC = 25 ft., DB
= 30 ft., LF AC =
20°, LBDE = 15°. Find LDBC and DC.
Ans. LDBC = 95°; DC = 40.69 ft.
49. ABCD is the ground plan of a barn of known dimensions AB
= a
and AD = b. A surveying party, wishing to locate a point P in the same
horizontal
Determine
.
plane with the barn, measure
the lengths
of the lines PB
the angles
= x,
DPC
=
a and
BPC
=
{3.
PC = y, and PD = z.
a sin (<I> + a)
cos (cf>- (3).
, y =
Ans . x = _b cos <1>.
, z = a sin cf>.
=
sin {3
sin a
sin {3
sin a '
a + b cot {3
and tan <I>=
where <I>= angle DCP.
b + a co t' a
50. The jib of a crane makes an angle of 35° with the vertical.
If the
crane swings through a right angle about its vertical axis, find the angle
between the first and the last positions of the jib.
Ans. 47° 51' 18".
51. If the jib of a crane makes an angle cf>with the vertical and swings
about the vertical axis through an angle 8, show that the angle a between the
first and last positions of the jib is given by the equation
_b
sin ~a = sin cf>sin !8.
l
OBLIQUE
PLANE AND SPHERICAL TRIGONOMETRY
156
consecutive
ribs
is given
by the equation
sin ~o
=
sin
~
sin
<1>.
63. To layout a pentagon in a circle, draw two perpendicular
diameters
AB and CD (Fig. 101) and bisect AD at E. With E as 'a center and ED
tan 1) =
D
B
A
C
FIG. 101.
FIG. 102.
as a radius, draw the arc DF. The length of the chord DF is the side of
the inscribed pentagon.
Prove this.
64. To layout
a regular heptagon in a circle, make a construction
as
shown in Fig. 102. AB is very nearly the side of the inscribed regular heptagon.
Determine the error in one side for a circle with a radius of 10 in.
and determine
the per cent of error.
Determine
the angle at the center
intercepting
the chord found by this
C process.
A ns. 0.2 per eel! L tuu small,
66. If the angle of slope of a plane is
0, find the angle of slope x of the line of
A
intersection
of this plane with a vertical
FIG. 103.
plane making an angle a with the vertical plane containing the line of greatest slope.
(Note the difference between
this exercise and Exercise 18, page 99.)
Suggestion.-In
Fig. 103, AD = a cot 0, AG = a cot x.
a cot 0,and tan x = tan 0 cos a.
. . . cos a = a cot x
66. Two vertical faces of rock at right angles to each other show sections
of a geological stratum which have dips (angles with the horizontal) of a and
f3 respectively.
If 1) is the true dip (angle between the stratum and a
horizontal plane), show that
tan2 1) = tan2 a + tan2 {3.
67. Two vertical planes at right angles to each other intersect a third
plane that is inclined at an unknown angle 0 to a horizontal plane.
If the
intersections
of the vertical planes with the third plane make angles of a
and {3,respectively,
with the horizontal plane, find the secant of o.
Ans. sec 0 = VI + tan2 a + tan2 (3.
157
Note.-The answer to the above is an important formula used in calculus.
58. To determine the dip of a stratum that is under ground, three holes
are bored at three angular points of a horizontal square of side a. The
depths at which the stratum is struck are, respectively, p, q, and r ft. Show
that the dip 1)of the stratum is given by the equation
62. An umbrella is partly open and has n straight ribs each inclined at
an angle <I>with the center stick of the umbrella.
Show that the angle 8
between
TRIANGLES
.
v(p
- q)2 + (q - r)2.
a
~
MISCELLANEOUS TRIGONOMETRIC
2a
That is,
or
(n
Check.-When
CHAPTER X
MISCELLANEOUS TRIGONOMETRIC
EQUATIONS
98. Types of equations.-In
this chapter equations of the
following types will be considered:
(1) Where there is one unknown angle involved in trigonometric functions.
(2) Where the unknown is not an angle but is involved in
inverse trigonometric functions.
(3) Where there are other unknowns, as well as unknown
angles, involved in simultaneous equations; but only the angle
involved trigonometrically.
(4) Where the unknown angle is involved both algebraically
and trigonometrically.
It is not possible to give general solutions of equations of all
these types. They offer algebraic as well as trigonometric difficulties.
Methods of solution are best shown by examples.
24
or 12 tan 2 0 + 7 tan 0
7'
.
-7 +
- V49 + 576 =
Solvmg for tan 0, tan 0 =
24
When tan 0 =
1, sin
0
-- 12 -
O.
-34 or 3:1'
= ! and cos 0 = t.
When tan 0 = -t, sin 0 = t and cos 0 = -I.
The student can easily verify these by triangles or formulas.
2. Given tan-I (a + 1) + tan-I (a - 1) = tan-I 2; find a.
Solution.-Let
0 = tan-I
(a
+
+a -
2 = 0,
1) = 0, whence a
= -2 or 1.
a = -2, tan-I (-1)
+ tan-I (-3)
-1-3
- (-1) ( -3)
= tan-II
Ans.
5. 2 cos2 20
6. sin 20
+
+ cos.20
+ sin
sin 40
Suggestion.-Apply
- 1 = 0; find o.
60 = 0; find o.
+ sin
[25] to sin 20
and equate each factor to zero.
7. cos 20 = sin 0; find o.
8. Given tan-I (a + 1) + tan-I
Ans.
:!:
VI3'
(11. :!: iJ7I", (211.
V!3'
1r
+ l):r
Ans. 11.;, (211.+ 1 :!: j)7I".
60.
Factor
Ans.
(211. + ! :!: j)7I", (211. + V 71".
the resulting
equation
(a - 1) = tan-I (-l,);
solve for a.
Ans. 7.137 or -0.280.
9. Given r sin 0 = 2 and r cos 0 = 4; solve for rand o.
Ans. r = 2,,/5; 0 = 26° 33' 53", 206° 33' 53".
Suggestion.-Square
both equations and add, to obtain r. Divide the first
by the second to obtain O.
10. Given tan-I (x + 1) + tan-I (x - 1) = tan-II;
find x.
Ans. 0.610 or -3.277.
Given
eu:s-l
(1 -
aj
+ (;u:s-l
a
=
(;u:s-1 (-a);
:sulve fur a.
Ans. 0 or !.
1. Given tan 20 = ~<-;find sin 0 and cos 0 without finding 0, for values
in the first and second quadrants.
2 tan 0
.
[21], tan 20 =
Solutwn.-By
. . . 1 2- tantan20 0 --
-
+ 2)(a
2, whence a2
159
-4
= tan-I -2 = tan-I 2.
3. Given sin 20 = 2 sin 0; find 0 < 360°.
Ans. 0, 71".
Suggestion.-Use sin 20 = 2 sin 0 cos 0 and factor.
4. Given tan 20 = Jl; find sin 0 and cos 0 for 0 in quadrants I and II.
2
3
11.
EXERCISES
2 - a2 =
EQUATIONS
1); then tan 0 = a
+
1.
Let {3= tan-I (a - 1); then tan {3= a - 1.
tan 0 +tan{3
a+l+a-1
2a
t an ( 0+ (3) = 1 - tan 0 tan {3= 1 - (a + l)(a - 1) = 2 - ai
2a
tan-I 2.
.'. 0 + {3= tan-I 2 - a2 =
158
12. Given r sin (0 - tan-I!)
Suggestion.-Obtain
by the other; expand
= 3, and r cos (0 - tan-I!)
= 6; find rand o.
Ans. r = 3V5; 0 = 40° 36' 5", 220° 36' 5".
r as in Exercise 9. To obtain 0, divide one equation
the functions and solve for tan O.
13. sin 40 = 2 cos 20; find o.
Am.
(11.
71"
+ !):l'
_
14. tan-l a+1
tan-l a-I.
= tan-I x. Fmd a when (a) x = 1,
a - 1+
a
(b) x = 2, (c) x
= -7.
Ans. (a) a = 0, (b) a = ! or -1, (c) a = 2.
Ans. (11.:!: V7l".
15. tan 2a tan a = j; find a.
16. sin (120° - x) - sin (120° + x) = !V3; find x.
Ans. 60°, 120°.
17. cos (30° + 0) - sin (60° + 0) = -!V3; find o. Ans. 60°, 120°.
18. VI - vsin< x + sin2 x = sin x-I;
solve for x.
19. V7 sin a - 5 + V 4 sin a-I
= V7 sin a - 4 +
solve for a.
20. Given tan (80° - !O) = cot ~O; find o.
21. Given 3 sin-I x + 2 cos-I x
= 240°; solve for x.
22. Given tan-I x + 2 cot-I x = 135°; solve forx.
V4
Ans.
sin a
-
(211.+ !)71".
Ans. 60°.
Ans.
2;
!V3.
Ans.1.
~
PLANE
160
AND
SPHERICAL
MISCELLANEOUS
TRIGONOMETRY
= 2 see' x - 6; solve for x.
Am. 45°, 225°, 116° 33' 56", 296° 33' 56".
24. Given 10 eos 8 - 5 sin 8 = 2; show that 8 = 2 tan-1 !.
known.
+5
Solution.-Either
COS
6
= i, for 6, when r, 5, and i are
sin (Jor cos (Jcan be eliminated by
100. Equations in the form
where !I, a, and ~ are variables.
equations and adding,
()
+
moos
(3)/n
'Y
cos (J = t, which,
Now m and
-
'Y
COSa
00
to
+
00,
by [161, gives
'Y)
From this (Jis determined.
'Y from (3).
Example.-Given
3 sin (J + 4 cos (J = 2; find (J.
3,
Solution.-This
is of the form given in this article, and r =
y9 + 16 = 5.
8 = 4, and t = 2. .'. m = yr2 + 82 =
0
tan'Y =
Since rand
sr = 3 = 0.75, and'Y
4:
8 are both
positive,
= 36° 52' 12" or 216° 52' 12".
sin
Therefore, 'Y is in the first quadrant,
only.
cos «(J- 'Y) =
and
~=
~
=
'Y
and
= b
Solution.-Squaring
cos 'Yare
positive.
-'
--I
, or a =
C
= ~=
P
Iya2
+ b2 + c2
=
=
=
a2
a2
a2
all three
+ b2 + c2.
+ b2 + c2.
+ b2 + c2.
(J - 'Y = 66° 25' 18" or 293° 34' 42".
. . . (J = 66° 25' 18" + 36° 52' 12" = 103° 17' 30",
(J = 293° 34' 42" + 36° 52' 12" = 330° 26' 54".
C
Iya2
+ (3) - c
sin a
-1'
sin (a
Taking the proportion
by composition and division,
sin (a + (3)+ sin a
c
+1
sin (a + (3)- sin a - c - (
By [25] and [26], 2 sin (a + t(3) cos t(3 = c + 1.
2 cos (a + t(3) sin tf3
c - 1
tan (a + t(3) c + 1
t an 2"
1(3' = c1'
1
1
tan a + 2(3 = c+l tan 2(3.
c- 1
A ppymg
I'
[7] ,
(
or
(3).
cos-1
.
+ b2 + c2
p sin a sin f3 b
. .
.
D IVIdmg
th e secon d equa t IOn b y th e firs,.t
'
p sm a COSR = -.
I-' a
b
b
Whence , tan f3 = -' f3 = tan-1 -.
a'
a
101. EquatioD.£ in the form sin (a +~) = Csino., where ~ and
c are known. Sulutiun.-Diviuillg
by ~ill a,
and so must be 36° 52' 12"
0.4, by equation
~.
From the third equation,
= t.
can be determined from (1) and (2), and then
cos (0 -
sin a sin
p2 sin2 a cos2 f3 + p2 sin2 a sin2 f3 + p2 cos2 a
p2 sin2 a (cos2 f3 + sin2 (3)
p2 cos2 a
+
p2(sin2 a + cos2 a) = p2
Then m is real if rand 8 are real quantities.
Dividing the first equation of (1) by the second,
r
tan 'Y = s'
(2)
'Y
!I
{ !I cos a = C
where m is a positive constant, and 'Y an auxiliary angle.
Such an assumption is always permissible, for, squaring both
equations of (1) and adding,
r2 + 82,orm = yr2 + 82.
m2sin2'Y +m2cos2'Y = r2 + 82,orm2 =
m sin
~= a
!I sin a cos
means of the relation sin2 (J + cos2 (J = 1, but logarithms are not
applicable to this solution.
A solution will now be given in which
the computations may be done by logarithms.
(1) Let
m sin 'Y = r, and m cos 'Y = 8,
Since the tangent may have any real value from when rand 8 are real, the angle 'Y will always exist.
Substituting (1) in the original equation,
161
The method given in this article enables one to combine two
simple harmonic motions of the same period into a single simple
harmonic motion of the same period.
Thus, r sin (J I 8 COS() becomes m cos «(J=+='Y).
23. Given tan 2x (tan' x-I)
99. To solve r sin 6
TRIGONOMETRIC EQUATIONS
)
From which, since Band c are known, a may be found.
Example.-Solve
sin (a
Solution.-Substituting
mula, w.e have
+
5'0°)
= 2 sin a.
50° for (3and 2 for c in the above for-
~
162
PLANE
AND SPHERICAL
TRIGONOMETRY
MISCELLANEOUS
tan (a + 25°) = 2+1 tan 25° = 3 tan 25°.
2- 1
log 3 = 0.47712
log tan 25° = 9.66867
log tan (a + 25°) = 0.14579
a + 25° = 54° 26' 29" or 234° 26' 29".
TRIGONOMETRIC EQUATIONS
163
the value of t where the line and the curve intersect.
The more
nearly accurate the sine curve is plotted the more nearly will the
value of t come to the solution of the equation.
Example.-Given
t = 2 + 11' sin t.
Let Yl = t - 2 and Y2 = 11'sin t.
Now plot Yl = t - 2, giving the line AB, as in Fig. 104. Also
. plot the modified sine curve with equation Y2 = 11' sin t.
a = 29° 26' 29" or 209° 26' 29".
y
102. Equations in the form tan (a + ~) = c tan a, where ~ and
c are known. Solution.-Dividing
by tan a and taking the
resulting proportion by composition and division,
+ (3) - c
-
tan (a
T
tan a
tan (a + (3)+ tan a
c+ 1
tan (a + (3) - tan a - c - (
sin (a + (3)
sin a
cos (a + (3)+ ~
+ (3)
sin (2a
c- 1
sin (a + (3) sin a - sin [(a + (3) - a] - c + (
cos (a + (3) - cos a
. . . sin (2a + (3) = cc- + 11 sin {3.
x
x'
Since c and {3are known, a may be found.
Example.-Given
tan (a + 24°) = 4 tan a; find a.
.",nll/tilm -~lJh"titlJtin/,:
mula, we have
sin (2a
+ 24°)
=
A
24° for {i Hnd 4 for r; in the abo~
y'
sin 24°.
log 5 = 0.69897
log sin 24° = 9.60931
colog 3 = 9.52288
log sin (2a + 24°) = 9.83116
2a + 24° = 42° 40.7', 137° 19.3', 402° 40.7', 497° 19.3'.
2a = 18° 40.7', 113° 19.3',378° 40.7', 473° 19.3'.
.'. a = 9° 20.4', 56° 39.6', 189° 20.4', 236° 39.6'.
103. Equations of the form t = 6
+
<I>
sin
t, where
6 and
<I>
given angles.-First
express (J and q, in radians if not already
given.
Then t must satisfy the relation t - (J = q, sin t.
Let
FIG. 104.
%
Yl = t -
(J
and
't/2
=
q,
By measurement, the abscissa t for the point P of intersection is
found to be 2.86 radians, or 164°. This is, therefore, an approximate solution for the equation.
Substituting for t in the original equation,
2.86
=
2
+
11'
sin 164° = 2
+
0.8659
=
2.8659.
This result shows the value of t to be too small.
Substituting t = 165°,2.88 = 2 + 11' sin 165° = 2.813.
This result shows that 165° is too large, which the intersection
of the curves also verifies. The correct value may now be
approximated by assuming values of t between 164 and 165°,
say 164° 10', etc.
are
so
sin t.
EXERCISES
1. Change 3 cos /I + 4 sin /I to the form m cos (II ,.),
Ans. 5 cos (II - 53° 7' 45").
Plot the straight line with equation Yl = t - (J,and the sine curve
Y2 = q, sin t. An approximate value of t can be determin,ed from
~
l
PLANE AND SPHERICAL TRIGONOMETRY
164
2. Change a cos ",t + b sin ",t to the form m cos (",t - ,,).
+
Ans. Va2
3. Given 5 sin
Suggestion.-sin
b2 COS
("'t - tan-'~)-
(J
- 2 cos = 3; find (J. Ans. 55° 39' 20",167° 56' 50".
is + and cos
will be in second
is -; therefore,
(J
"
"
quadrant.
4. Given 2 sin (J + 5 cos
(J
6. Given 1.31 sin
(J
= -3;
- 3.58 cos
(J
"
CHAPTER
find (J.
Ans. 145° 39' 20", 257° 56' 50".
= 1.885; find (J.
COMPLEX
Ans. 99° 32' 10", 220° 16'.
Suggestion.-Use
logarithms in the solution.
p sin a cos {J = 3,
p sin a sin {J 2,
=
p cos a = 1; find a, {J,and p.
Ans. p = VI4; a = 74° 29' 56"; {J = 33° 41' 24".
6. Given
7. Given
p sin (J cos </>=
p sin
(J sin
</>
6,
= 2,
p cos (J
= 0; find (J, </>,and p.
p
Ans.
= 2.y'IO; (J = 90°; cf>= 18° 26' 5".
8. Given sin (x + 32° 16') = 4 sin x; find x.
Ans. 9° 36' 23", 189° 36' 23".
9. Given sin (y - 75°) = 3 sin y; find y. Ans. 160° 35.3', 340° 35.3'.
10.
Given
tan
(r
+
40°) = 5 tan r; find r.
Ans. 17° 18.6', 32° 41.4', 197° 18.6', 212° 41.4.
11. Given tan (s - 60° 20') = 2 tan s; find s.
12. Given x = 1 + 30° sin x; find x approximately.
13. Given S = 60° + sin S; find S approximately.
§
..-
NUMBERS,
XI
DEMOIVRE'S
THEOREM,
SERIES
104. Imaginary numbers.-If
the equation x2 + 1 = 0 is
solved, we obtain the symbol y=I,
and from its derivation the
~
square of this symbol must be
1.
The symbol v=1 is commonly represented by i, and is called the unit of imaginaries.
It follows that, if i is a number, it is such a number that i2 = -1.
This is taken as the definition of i.
As the only property attached to i by its definition is that its
square is -1, it may be multiplied by any real number a. The
product ai is called an imaginary number.
In contradistinction
to imaginary numbers, the rational
and irrational numbers, including positive and negative integers
and fractions, are called real numbers.
The name "imaginary number" suggests an unreality that does
not exist, for in the present state of the development of mathematics the imaginary number in comparison with whole numbers
is no more unreal in the ordinary sense than is the fraction or the
irratIOnal number.
The system of numbers created by accepting the imaginary
unit is an entirely new system of numbers distinct from the
system of real numbers.
The operations upon these numbers
and their combinations with real numbers present various applications of trigonometry;
and, further, the method developed
in this field can be used to advantage in deriving various formulas
in trigonometry.
105. Square root of a negative number.-By
definition, the
square root of a number is such a number that, when multiplied
by itself, will give the original number.
If c is a positive real
number, no real number can equal V=C, for, by the definition
of square root,
.y=c . V -c = -c.
But the square of every real number is positive.
Hence, the
square root of a negative real number is not a real number.
That
it is an imaginary number can be shown as follows:
165
m
-
---
-----..-----
--- ---------
-----
--------------------..------------
i2
-1,
=
VC
vei'
Therefore,
of
Art.
by
definition
=
VCi
=
.'. VCi
c,
of
c( -1)
square
=
-c,
-V=C,
taking
-v=c
is an imaginary
multiplying.
complex
root of each
It follows
that
any
called the proper
imaginary
number
number,
of the form
being of the form
..;=c,
can be put in the form
imaginary
number
form.
where
vei,
It is always
in the proper form
which
c is a
will be
vi=4
y=6
106.
that,
they
the
act
proper
like real
exception
.y4i
=
V6i.
=
an
2i.
with imaginary numbers.-It
Operations
with
=
definitions
numbers
of the
for
and
combining
obey
laws
numbers,
of algebra,
with
law
..;avrz;
=
yIOl).
This law is excepted because it conflicts with the definition of
the unit of
imaginaries, and a definition
apply, we should
Thus, if this hnv did
V-IV-l
is always fundamental.
.
= V(-I)(-I) = VI = 1,
a
+
b = 0, and an imaginary
bi where
number
.
I
B'
~
Y'
FIG. 105.
was discovered independently by
Wessel, a Norwegian, in 1797; by Argand, a Frenchman, in
1806; and by Gauss, a German, in 1831. More recently the
practical importance of complex numbers as graphically represented has been recognized by physicists and engineers. In
the field of electricity they have been put to important uses
by Steinmetz and others. The system for representing these
numbers is as follows:
Draw the rectangular coordinate axes X'OX and Y'OY (Fig.
105). The real numbers can be represented by points on the
line X'OX as follows: The point A, corresponding to the positive
number a, is taken a units to the right of O. The point A',
corresponding to the negative number -a, is taken a units to
the left of O. The number a can also be considered as represented
by the line segment OA, and the number -a by the line segment
OA'.
V -2V -3 = V( -2)( -3) = ..;6.
If the imaginary numbers are first put in the proper form, no
trouble will occur.
= V2i . y'3i = V6i2 = - V6-
107. Complex numbers.-In
order to solve the general quad0,
it is necessary to have numbers
ratic equation mx2 + nx + l =
bi.
These
numbers
are formed by adding a
of the form a +
real number to an imaginary number.
gives the two values
Thus, the solution of x2 - 4x + 13 = °
for x, 2 + 3i and 2 - 3i.
---------
l
number
introduced
by Descartes
(15961(50).
The system of representing complex numbers graphically
have
whereas, by definition, V=I -FI
= i2 = -1.
It, therefore, contradicts the definition of the unit of imaginaries to say
Thus, yC2yC3
167
109. Graphical representation
of complex numbers.-Long
after imaginary numbers presented themselves in algebraic work,
they were rejected by mathematicians
as impossible.
Indeed,
this was the case with any new
y
kind of number. The negative
+B
n urnber was disregarded for
:c
centuries after it appeared, and
was generally accepted only after x.!. -a
x
a
0
A"
A
its graphical representation wa s
:.c
can be shown
imaginary
all the
SERIES
EXERCISES
Write the complex numbers conjugate to the following:
3. 1 + V3i.
5. -7 - -yI-=I. 7. V6i.
1. 2 - 3i.
6.3y.=-[
8. v=-r
2.3 + i.
4. V2 - y=3.
best to write an
before performing
operation.
Thus,
THEOREM,
is a special case where a = 0.
108. Conjugate complex numbers are complex numbers which
differ only in the signs of their imaginary parts. Thus, 3 - 4i
and 3 + 4i are conjugate complex numbers.
Also +2i and -2i
are conjugate.
member.
104.
positive real number
DEMOIVRE'S
and i2
root.
by
the square
NUMBERS,
Numbers oftheform a + bi, where a and b are real numbers
= - 1, are called complex numbers.
It should be noted that a real number is a special case of the
by definition of unit of imaginaries.
VC =
.
Then
ai
COMPLEX
PLANE AND SPHERICAL TRIGONOMETRY
166
~
~
PLANE
168
AND
SPHERICAL
COMPLEX NUMBERS, DEMOIVRE'S THEOREM, SERIES
TRIGONOMETRY
110. Powers of i.-It
power of i.
If the point A represents the number 5, the point A', representing the number -5, can be obtained by rotating OA through
180°. It seems, then, that multiplying a number by -1 acts
as if the multiplication rotated the line segment representing the
number through 180°.
Multiplying a number twice by i, as 5ii, is the same as multiplying it by -1.
In other words, it rotates the line segment,
representing the number 5, through 180°, but does not change
y
the length of the line segment. This
(3,,4) suggests that multiplying a number by
(-3.4)
~
i should rotate the line segment, repre~
senting the number, through 90°. Then
an imaginary number like bi would be
-f--X represented by a point B on the positive
x'.
0
y-axis and b units above the x-axis.
Likewise, the imaginary number - bi
p.
P3
would be represented by a point B' on
(3.L)
(-3.-4)
the negative y-axis and b units below the
Y'
FIG.106
x-axis. The imaginary number bi can
also be considered as represented by the line segment DB, and the
imaginary number -bi, by the line segment DB'.
The line on which the real numbers are represented is called the
axis of reals. The line on which the imaginary numbers are
represented IS called the axis uf illlaginaries.
By this system it is easy to represent complex numbers graphically. In Fig. 106, the complex number 3 + 4i is represented
by the point (3,4),3 units to the right of the y-axis and 4 units
above the x-axis. Likewise, the complex number 3 - 4i is
4i
represented by the point (3, -4); the complex number -3 +
by the point (-3,4);
and the compl~x number -3 - 4i by the
point (-3, -4).
The figure on which the complex numbers are plotted is called
the Argand diagram.
is a very easy matter
169
to compute any
il = i, by the definition of an exponent.
i2 = -1, by the definition of the unit of imaginaries.
= i2i = (-I)i = -i.
i4 = (i2)2 = (-1)2 = 1.
i5 = (i2)2i = (-1)2i = i.
i6 = (i2)3 = (- 1)3 = -1.
i3
.
0
By continuing this process, it is found that the integral powers
of i recur in a cycle of the four different
values i, -1, -i, and 1.
Y
B+i
The powers of i can, perhaps, be made
clearer by referring to Fig. 107, where each
~
+
multiplication by i rotates the line segment
."I,j/
+
' \~.
-1
x
of unit length through 90°.
-+
.
C ,\
L/ +\.
.
A
Th e va 1ue 0 f any IIIt egra 1 power 0 f t can
""':'~-.J">.
~
be readily found as illustrated in the
following examples:
D+-i
~-'~
0
.
il7
FIG. 107.
=-1.
EXERCISES
Compute the following powers of i:
1. i7.
6. i20.
11. (-i)'o.
2. i9.
7. i26.
12. (-i)l1.
3. i13.
8. i30.
13. -i6.
4. i17.
9. i36.
14. -i8.
5. i18.
10. (-i)8.
15. i'o,.
16.
17.
18.
19.
20.
il2°.
i202.
-i300.
-i'ooo.
(-i)667.
111. Operations on complex numbers.-Complex
numbers,
under proper definitions for the four fundamental operations,
obey all the laws of algebra, with the exception of the law mentioned in Art. 106. In fact, complex numbers act the same as
real numbers.
The four fundamental operations are defined as follows:
Addition.
(a + bi) + (c + di) = (a + c) + (b + d)i.
EXERCISES
Plot the following
= (i2)8i = (-1)8i = i.
il8 = (i2)9 = (-1)9
...
complex numbers:
1. 2 + 3i; -2 + 3i; 2 - 3i; -2 - 3i.
2. 4 - i; 4 + i; 6i; -5i; -6 + 3i; -3 - 5i; 4.
3. ... + i; ". + 1fi; -1 - 1fi; e + 1fi; ". - ei; e - i; -1 - ei.
4. i; i2; i3; i4; i5; i6; i7; is.
Plot the following complex numbers and their conjugates:
5. 1 - i, 2 + i; 2i; 3 - 2i; 2i - 4.
Subtraction.
Multiplication.
(a
+
(a
bi)
+
- (c + di) = (a - c)
bi)(c + di) = ac + adi
+
+
(b - d)i.
bci + bdi2
= ac - bd + (ad + bc)i.
.&
l
81
170
PLANE
AND
SPHERICAL
(a +
. ..
a + bi
D IVlslOn.
c + di - (c +
ac +
= c2
TRIGONOMETRY
COMPLEX
bi) (c - di)
ac + bd + (bc - ad)i
c2 + d2
di) (c - di) bd (bc - ad)i.
+ c2 + d2
+ d2
Note that division cannot be defined if c2 + d2 = 0, that is,
c = 0 and d = O. In this case it will be seen in the next article
that
the
complex
number
c
+
di = O.
Hence,
in the
complex numbers, division by zero is impossible.
Note also that the four fundamental operations
numbers always yield complex numbers.
field of
on complex
EXERCISES *
Simplify the following by performing the operations indicated:
17. (1 + y5i) + V5i.
1. (3 + 2i) + (6 - 7i).
18. 1 + (1 - i).
2. (1 + i) - (3 + 4i).
19. (1 - i)2.
3. (7 - i) + (3 + 4i).
4. (1 - 6i) - (-7 +
3i).
6. (3 + 2i) (1 + 5i).
6. (-3 + 2i)( -3 - i).
7.
4 - i
3.
'3+4i
10. ~.
1 - 2~
11. 0 + ~i.
1 - y2i
13.
28. (i9 +
~i.
6 + 3i
2 + 3i
.
.y2 - v'3i
5
.
14.
0 + V3i
7i,
15. 6
~
16. 1 ti 3i.
i10
+ is + i9)
30.
+ i9 + i12)6.
(i6
31. (2a-2 -
32. (0
Ar.
33. Prove that 1 + i is a root of the equation
2x3 - x2 - 2x + 6 = O.
34. Find the value of x3 - 2X2 + 9x + 13 when x = 2
SERIES
171
. the value of (x2 5X)2 x(x +5) when x = -5 + iV3 .
35. Fmd
+ +
36.
Find
the value
of
.
37. Fmd the value of
3X2 - 4x + 12
x2
+
x
+
1
when
x
=
3
+
Ans.2.
i.
- 6x + 9
.
Ans. 2.
2X2 - x + 2 when x = 2 + ~.
38. Prove that the sum and the product of two conjugate complex numbers are both real.
39. Prove that, if the sum and the product of two complex numbers are
real, the numbers are conjugate complex numbers.
5X2
112. Properties of complex numbers.
THEOREM I.-If
the
complex number a + bi = 0, then a = 0 and b = O.
Proof.-Since
the laws of algebra for real numbers hold with
one exception for complex numbers, if bi is transposed to the
right-hand side, then a = - bi. Squaring both sides of this
equation gives a2 = -b2. But a positive number cannot equal
a negative number unless both are zero. Hence a = 0 and b = O.
THEOREM II.-If
a + bi and c + di are two complex numbers
such that a + bi = c + di, then a = c and b = d.
Proof.-If
equation, a
c
+
di is transposed
- c + (b -
d)i
= O.
to the left-hand
side of the
Then, by Theorem I, a
-
c= 0
EXERCISES
Find the real values of x and y for which the following equations are true.
1. x + y + (2x + 3y)i = 3 + i.
Ans. x = 8, y = -5.
Suggestion.-Apply
Theorem II, which gives
x + y = 3, and 2x + 3y = 1.
2. 3x - 2 +
12.
+ ~)
THEOREM,
The proof of this theorem is to be given as an exerciRe.
+ ill + i12)7.
29. (i7
DEMOIVRE'S
and b - d = O. Hence, a = c and b = d.
THEOREMIlL-If the product of two complex numbers vanishes,
at least one of the factors must vanish, and conversely.
y
26. v=a + -'-b
v=a-a - v-b
(),
26. + i v/:L7.
a - i~
27. a + iVf+ll2.
a - iYl - a2
9. 6_7-- 3i~.'
3 -
22. (Y3 - y -2)3.
3
-1 +~
.
23.
( 2 )
1 + v=3
24 . (
)
2
2+{
5 + 2i.
8
12.
20. (1 + i)3.
21. (3 - 4i)2.
NUMBERS,
3.
+ 3i.
* Answers to above problems will be found at the end of this chapter.
(-2)i
=
y(1
- i).
Ans. x = Ii; y = 2.
y + 16 + 2(y + l)i = 2x(2 - i).
Ans. x = 3; y = -4.
x(1 + i) + y(1 - i) = 2.
Ans. x = 1; y = 1.
x2 + 2xyi + y2 = 25 + 24i. Am. x = ::!:4 and::!: 3; y = ::!:3 and::!: 4.
x(x + i) + y(y + i) = 5 + 3i.
Ans. x = 1 and 2; y = 2 and 1.
The product of two complex numbers is 5 - i, the sum of their real
is 3, and the product of their real parts is 2. Find the numbers.
Ans. 1 + i and 2 - 3i, or 1 - !i and 2 + 2i.
8. Find two conjugate complex numbers whose product is 13, and the
product of whose iIr.aginary parts is 9.
Am. 2 + 3i and 2 - 3i.
3.
4.
5.
6.
7.
parts
113. Complex numbers and vectors.-By
means of the Argand
diagram the general complex number a + bi is represented by
the point P (Fig. 108), with coordinates (a, b). In Art. 109,
l
172
COMPLEX NUMBERS, DEMOIVRE'S THEOREM, SERIES
PLANE AND SPHERICAL TRIGONOMETRY
The values of rand 0 in terms of a and b can be obtained
immediately from Fig. 108. They are
it was seen that a real number or a pure imaginary number can
be represented by either a point or a line segment.
This notion
can be extended to complex numbers by representing the complex number a + bi by the line segment OP. For, if the segment
OP is given, a, the real part of the complex number, equals the
projection of OP on the x-axis, and b, the coefficient of i, equals
the projection of OP on the y-axis.
Definition.-A
quantity that has magnitude as well as direction
is called a vector.
The line segment OP begins at the origin and ends at P. It,
therefore, has a magnitude and a direction.
Hence the complex
number P can be represented by the vector OP. Hereafter, the
word" vector" will be used in place of "line segment."
114. Polar form of complex numbers.-The
vectorial representation of a complex number enables it to be written in another
form, called the" polar form" of the complex number. Let
IY
0 be .the angle through
. which the positive
P(a+bi)
por t lOn 0 f t h ex-aXIS wou ld h ave to b e
T=
V d~ + b2, and 6 = sin-l
Example I.-Write
A-- X called the amplitude,
complex number
FIG. 108.
a
or argument,
+ bi.
+
y
in the polar form.
Plot.
2V3i
=
+
= 4(cos 60°
= 60°.
i sin 60°).
2+2\1'3i
x
~
t;
~
I
~
X
2
-2-2'V3i
(a)
of the
Let r be the
(b)
FIG. 109.
1'he plotting if: shO'.m in Fig. 109"
Example 2.-Write
-2 - 2V3i in the polar form.
r = '\)(_2)2
()
+(-
2Y3)2
Plot.
= 4.
-2V3
~ /<)
= tan-1 ~
= tan-l v 3 = 240°.
Here 240° is taken because both a and b are negative and 0
is, therefore, in the third quadrant.
i sin 6).
The expression r( cos 0 + i sin 0) is called the polar form of
a complex number.
The expression a + bi is called the rectangular form of a
complex number.
If 0 is increased, or decreased, by multiples of 360°, the sine
and the cosine are not changed, then the polar form of a complex number can be written
k. 360°) + i sin (0 + k . 360°)],
r[cos (0
.'. -2 - 20i
= 4(cos 240° + i sin 240°).
The plotting is shown in Fig. 109b.
Note that, while a and b may be negative numbers, r is always
positive, and that the signs in front of cos 0 and sin 0 are always
plus.
The complete palar forms of the complex numbers in the preceding examples are
2 + 20i
= 4[cos (60° + k . 360°) + i sin (60° + k .
360°)],
+
where k is any positive or negative integer.
complete polar form of a complex number.
+
.'. 2
whose sides are a, b, and r,
a = T COS6, and b = r sin 6.
Note that these equations hold no matter in what quadrant
0 lies. The complex number a + bi can now be written
bi = r(cos 6
T
+
~
+
+
+ 20i
= COS-l ~r = tan-1 !!..
a
!!.
(2Y3)2 = 4.
0 = tan-1 20
tan-1 ~r
v 3
length, or magnitude,
of the vector OP.
The numberr
is called the modulus
of the complex number
bi, and is always taken positive. From the right triangle
a
a
2
r = '\)22
revolved
in order to coincide with the
vector OP (Fig. 108).
The angle 0 is
b
173
This is called the
~
~
174
PLANE
AND
SPHERICAL
and
COMPLEX
TRIGONOMETRY
.
i sin (240° + k 360°)].
+
-2 - 2V3i = 4(cos (240° +
Suggestion.-In
changing from rectangular form to polar form,
it is better first to plot the complex number, and then check the
results obtained by computation with those indicated on the
graph. Thus, the common error of writing for 8 a first-quadrant
angle when 8 is an angle in some other quadrant may be avoided.
Often the values of rand 8 can be obtained directly from the
figure.
k . 360°)
EXERCISES*
Write the following complex numbers in the polar form:
1. 1 + i.
Ans. 0 (cos 45°
2. -1 + i.
Ans. 0
3. - 1 - i.
4. 1 - i.
5. - V3 +
Ans. 0 (cos 225°
Ans. V2 (cos 315°
6.
7.
8.
13.
14.
15.
3i.
-v'3 - 3i.
v'6 + 30i.
-0
+ 0i.
4.
4i.
-6.
16. -6i.
17. i.
9. -30
18. -i.
19. 1.
20.
+
+i
+ 3i.
35. - v'3 - 3i.
1
+ 3i.
+
i sin 60°.
36. -1 + V3i.
37. 3i.
Write the following complex numbers in the rectangular form:
38. 3 (cos 30° + i sin 30°).
46. 2 (cos 150° + i sin 150°).
39. 0 (cos 45° + i sin 45°).
47. 2 (cos 510° + i sin 510°).
40. 4(cos !11' + i sin !11').
48. cos (- 210°) + i sin ( - 210°).
41. 2 (cos !11'+ i sin !11').
49. cos (-570°) + i sin (-570°).
42. 4 (cos t1l' + i sin ~).
50. cos 100° + i sin 100°.
43. 4 (cos ~11'+ i sin ~11').
51. 2 (cos 200° + i sin 200°).
44.
10 (cos !11'
45. 6 (cos 720°
+ i sin !11').
+ i sin 720°).
i sin 300°).
52. 3 (cos 300° +
53. 4 (cos 1000° + i sin 1000°).
115. Graphical representation
of addition.-Let
the vector
OP (Fig. 110), represent the complex number a + bi, and let
the vector OS represent the complex number c + di.
Answers to some of the above problems
*
chapter.
175
In order to represent graphically the sum of a + bi and c + di,
complete the parallelogram OPTS by drawing PT parallel to
OS and ST parallel to OP. Then the vector OT represents the
complex
number
(a
+
bi)
+
(c
+
di).
Proof.-Drop
perpendiculars from P and T to the x-axis, and
call the feet of these perpendiculars Pl and Tl' respectively.
Also drop perpendiculars from Sand T to the y-axis, and call
the feet of these perpendiculars S2 and T2, respectively.
Then the real part of the complex number represented by the
vector OT is OTl = OPl + PlTl = a + c.
x
sin 315°).
29. -cos 75° + i sin 75°.
32. sin 30° + i sin 240°.
30. -3(cos 10° + i sin 10°).
33. -sin 210° - i sin 120°.
Write the following complex numbers in the complete polar form:
34. 3
SERIES
i sin 225°).
26. 4 - 2i.
2'1. -4 + 3i.
31. cos 30°
THEOREM,
- 3V6i.
25.
21. 0.
22. V -3.
DEMOIVRE'S
+ i sin 45°).
+ i sin 135°).
10. 0 + V6i.
11. V5 + V15i.
12. -vIzl + v7i.
23. 2i2.
24. 3i3.
-1.
28. cos 30° - i sin 30°.
(cos 135°
NUMBERS,
will be found at the end of this
x
s~
-c-di
FIG. 110.
FIG.11i.
The coefficient of i for the complex number
the vector OT is OT2 = OS2 + S2T2 = b + d.
ThPrefore,
()Trepresents
the complex
nnmher
represented
a
+ +
('
(h
by
+ d)i.
116. Graphical representation
of subtraction.In order to
represent graphically (a + bi) - (c + di), write the expression
in the form (a + bi) + (-c - di). In Fig. 111, produce the line
OS through the origin to a point S', so that S'O = OS. The
vector OS' represents the complex number -c - di. Then add
the vector OS' to OP precisely as was done in the case of addition.
EXERCISES
Perform the
algebraically:
1. (3 4i)
+
2. (-3
following
operations
+ (5 + 2i).
+ 2i) + (6 - 3i).
3. (2 + i) + i.
4. 2 + (3 - i).
graphically,
and
check
the
results
5. (3 + 4i) - (5 + 2i).
6. (-3 + 2i) - (8 - 3i).
7. (2 + i) - i.
8. (1 - i) + (-2 + 3i) + (4 + i).
9. 2 (cos 45° + i sin 45°) + (cos 135° + i sin 135°).
10. (cos 30°
11. (cos 0°
12.
(cos 60°
+ i sin 30°) + (cos 60° + i sin 60°).
+ i sin 0°) + 4 (cos 90° + i sin 90°).
+ i sin
60°) +(cos
180° + i sin 180°) +(cos
300° + i sin 300°).
13. 2 (cos 120° + i sin 120°) - 3 (cos 135° + i sin 135°).
l
PLANE AND SPHERICAL TRIGONOMETRY
176
COMPLEX NUMBERS, DEMOIVRE'S THEOREM, SERIES
117. Multiplication
of complex numbers
in polar form.
THEOREM.-The modulus of the product of two complex numbers
equals the product of their moduli, and the amplitude of their
product equals the sum of their amplitudes.
Proof.-Let
r1(cos Ih + i sin 81), and 1'2 (cos 82 + i sin 82) be
two complex numbers.
Then their product is
+i
and the amplitude of the quotient equals the amplitude of the dividend
mi1/:us the amplitude of the divisor.
Proof.r1(Cos 81
r2(cos 82
81
sin 81)]h(cos
82
+ i sin
Example.-[5(cos
82
15°
i sin
15°)][6(cos
20°
+ i sin 35°).
If the result is required in the rectangular
cos 35° and sin 35° in trigonometric tables.
+
i sin
Example.-16(cos
form, find the value
,y
sides of similar
OP3
OP1
- OA'
OP2
Whence
1'3
-
1'2
triangles
1'1
= -,1 or
are in proportion,
1'3
=
1'21'1.
Therefore,
r3(Cos
83
+
i sin 83)
=
r1r2[cos
(81
+
82)
+
i sin (81
+
82)].
119. Division of complex numbers in polar form. THEOREM.
The modulus of the quotient of two complex numbers equals the
modulus of the dividend divided by the modulus of the divisor,
81
+
i sin 81)(cos
+
+
+
82
-
i sin 82)
157°
+ i sin
157°) + 8(cos 22°
+ i sin
22°)
cally the complex number represented by the vector OP
3 by the
complex number represented by the vector OP1 (Fig. 112), construct a triangle OP?2 similar to the triangle OP1A and similarly
situated.
Then the complex number represented by the vector
OP2 is the required quotient.
The proof is left as an exercise.
= 30(cos 35°
Since corresponding
r1(cos
r2(cos 82
i sin 82)(cos 82 - i sin 82)
sin 81 sin 82)
i(sin 81 cos 82 - COS 81 sin 82)
cos2 82
sin2 82
= 2(cos 135° + i sin 135°) = - y2 + y2i.
120. Graphical representation of division.-To divide graphi-
20°)]
Thus, 30(cos 35° + i sin 35°) =
24.575 + 17.207i.
118. G rap h i c a 1 representation
of
multiplication.-Let
the vectors OP1
and OP2 (Fig. 112) represent the com~
plex numbers r1(cos 81 +i sin 81) and
Let
X r2(cos 82 + i sin 82), respectively.
the point A on the positive axis of reals
be 1 unit distant from the origin. Join
FIG.112.
PI to A
Construct a triangle OP?3
similar to the triangle OAP1 and similarly situated.
Then the
vector OP3 represents r3(cos 83 + i sin 83), the required product.
Proof.-83 = LAOP2 + LP2OP3 = LAOP2 + LAOP1 = 82 + 81.
+
-
1'2
+ i sin 82)] . . . [rn(cos 8n + i sin 8n)]
+ . . . 8n) + i sin (81 + 82 + . . . 8n)].
+
i sin 82)
= ~[cos (81 - 82) + i sin (81 - 82)].
81)][r2(cos 82
= 1'11'2'. . rn[cos (81 +
i sin 81)
1'1 (cos 81 COS 82
It is evident that this theorem can be generalized to include
the product of any number of complex numbers.
Thus,
[r1(cos 81
+
+
-- r;
+i
sin 82)]
i(sin
81cos
82 + COS81 sin 82)]
= r1r2[(cos 81COS82 - sin 81sin 82) +
r1r2[cos
(81
+
82)
+
i
sin
(81
+
82)].
By
[13] and [14].
=
h(cos
177
EXERCISES
iilii
Perform the following operations
form:
1. [3(cos 15° + i sin I5°)][6(cos
2. [4(cos I27°+i sin I27°)][3(cos
3. [2(cos §1T+ i sin ~1T)][5(cos b
4. [3 (cos 1'21T+ i sin l21T)][ 5 (cos
4(cos 47° + i sin 47°).
6
. 2(cos
and express
75°
+
203°+
the results
i sin 75°)J.
i sin 203°)1.
+ i sin ,201T)J.
+
15J!1T i sin lJ!1T)J.
in rectangular
Ans. I8i.
Ans.6v3
- 6i.
Ans. 5 + 5v'3'i.
Ans. I5i.
va
Ans.
+ i.
17° + i sin 17°)
12 (cos 26° + i sin 26°)
6.
Am. -4.
3(cos 206° + i sin 206°)'
Perform the following multiplications and divisions graphically and check
the results algebraically:
7. (1 + i)(2 + 3i).
9. (2 + i)(1 - 2i).
8. (10 + Hi) -7-(4 + i).
10. (3i - 1) -7-(1 + i).
11. Plot a + bi and i(a + bi). Does multiplying a complex number by i
rotate the vector representing the complex number through gOO? Prove.
121. Involution of complex numbers.-If
all the factors of
the generalized theorem for multiplication of complex numbers
(Art. 117) are equal, the result is the nth power of
l' (cos 8 + i sin 8).
Hence [r(cos 8 + i sin 8)]n = rn(cos n8 + i sin n8).
This result is known as DeMoivre's Theorem, discovered by
Abraham DeMoivre (1667-1754), who was French by birth but
lived in England after the age of seventeen.
l
COMPLEX
PLANE AND SPHERICAL TRIGONOMETRY
178
-
Note that the theorem has been established when n is a positive
integer only.
Example
fJ + i sin 0»)3
1.-[r(cos
Example 2.-[2(cos
40°
=
+ i sin 40°)]6
= 64( -!
r3(cos 30
=
+
+ i sin 240°)
- !V3i)
4.
5.
6.
7.
[2(cos
(h/2
(!V3
(hl2
+
i sin 120°)]'.
-8
+
Ans.
8V3i.
-16.
211 +
311 +
411 +
511 +
i
i
i
i
sin
sin
sin
sin
211 =
311 =
411 =
511 =
10.
COS '-)Ii -I_I. Kin 1)11 =
16. Using the results
17. Using the results
(cos
(cos
(cos
(cos
II +
II +
II +
II +
i
i
i
i
sin
sin
sin
sin
of Exercise
of Exercise
of tan II.
of tan II.
+ i sin o)]n = [r(cos 0 + i sin O)]q
[r(cos qrp + i sin qrp)]q = [r( cos rp + i sin rp)q]q
p
= ,q(cos
= rn(cos
1
= {V2[cos (315° + k. 360°) + isin (315° + k. 3600)]}3
. . 315° + k . 360°
315° + k . 360°
- V-6/ 2- cos
+ ~ sm
(
[r(cos 0
=
2k7l'
By Tn is meant the arithmetical value of -\Yr. Giving k the
values 0, 1, 2, . . . n - 1, in succession, all the nth roots of
the complex number can be found.
Example 1. Find all the cube lOot" of 1
i in poial fUllll.Plot.
Solution.-Plot
1 - i and find rand fJby inspection or computation.
1 - i = V2[cos (315° + k . 360°) + i sin (315° + k . 360°)].
~
p
p
. .
+ ~ sm
fJ
2k7l'
122. DeMoivre's theorem for negative and fractional exponents.-DeMoivre's
theorem is also true when n is not a positive
integer.
CASE I.-When
n is a positive rational number.
Then
o)]m = rm(cos mO + i sin mfJ)
i sin 0°)
I
~
----
12, express tan 311in terms
13, express tan 411in terms
and let 0 = qrp.
Let n = E
q
sin o)]-m
1
1
11)2.
11)3.
II)'.
11)6.
(('OK I) --r-'~ Kill 1))°.
179
+ i sin fJ) = [r(cos fJ + i.Bin O)]n
1
= {r[cos (0 + 2k1r) + i sin (fJ + 2k1r)]}n
!
0+
. . fJ +
= rn (cos
+ ~ SIn n ) .
n
V"r(cos
Ans. -! - !V3i.
cos
cos
cos
cos
+i
SERIES
(-mO)]
= rn(cos nO + i sin nfJ).
Hence DeMoivre's theorem is true when n is any rational
number.
The theorem can also be proved for irrational values
of n.
123. Evolution of complex numbers.-By
means of DeMoivre's
theorem, all the nth roots of a complex number can be found. It
will be seen that, in order to get all the roots, it will be necessary
to use the complete polar form of the complex number.
Thus,
Ans. -! + !V3i.
9. (-! + !V3i)lOOO.
Ans. 1.
10. [!VZ( -1 - i)]200.
In Exercises 11 to 15, raise the right-hand side to the indicated power by
the binomial theorem, simplify, and then apply Theorem II of Art. 112.
11.
12.
13.
14.
+
[r(cos 0
"
Ans. -1.
Ans. -h/3
- !i.
Ans. -1.
8. aV3 + !i)600.
+ i sin
=
THEOREM,
- rm(cos m fJ + ~ SIn m 0) -- r-m [cos (-mO)
= -32 - 32V3i.
Ans.
315° + i sin 315°)]'.
+ h/Zi)'.
- !i)5.
- h/Zi)lOO.
1
1 (cos 0°
EXERCISES
Simplify the following and express the results in the rectangular form
Plot in Exercises 1 to 6.
1. [3(cos 15° + i sin 15°)]6.
Ans. 729i.
2. [2(cos 50° + i sin 50)]6.
Ans. 32 - 32V3i.
3. [2(cos 120°
DEMOIVRE'S
Then [r(cos fJ + i sin fJ)]n
= [r(cos 0
i sin 30).
26(COS 240°
NUMBERS,
3
= {/2[cos (105°
p
3
+ k . 120°) + i sin
(105°
+k
)
. 120°)].
Giving k in succession the values 0, 1, 2, the required cube
roots are found. Representing them by Zl, Z2, and Z3, they are
p
+ i sin rp)p = rq(cos prp + i sin prp)
Eo+
i sin EO) = r"(cos nfJ + i sin nO).
q
q
rp
=
+ i sin
Z2
{/2(cos 105°
= {/2(cos 225°
Za
=
+i
+i
Zl
CASEII.- When n is a nega.tiverational number.
Let n = - m, where m is a positive integer or fraction.
.~
{/2(cos
345°
105°),
sin 225°),
sin 345°).
l
180
PLANE AND SPHERICAL TRIGONOMETRY
COMPLEX
These are all the cube roots, for, if k should be given values
greater than 2, no new cube roots would be found, as every root
so found would be either Zl, Z2, or Za. The plotting is shown in
Fig. 113.
The three cube roots can be changed to the rectangular form
by using logarithms to express approximately in decimals the
products indicated.
This gives the following:
=
Zl
-0.2905
+
Z3
Then
y
-1
Zl
y
Z2
Z3
Z4
Z3
FIG. 114.
Example 2.-Find
all the cube roots of 1 in rectangular form.
Plot.
Solution. The modulus of 1 is 1 and ib amplitude is 0".
Hencevi
= [cos (k . 360°)
= cos (k . 120°)
~
+ i sin (k . 3600)]~
+ i sin
(k' 120°).
Giving k in succession the values 0, 1,2, the three cube roots of
1 are as follows:
Zl = cos 0° + i sin 0°.
Z2 = cos 120° + i sin 120°.
Z3
= cos 240°
+
Changing these to the rectangular
Zl
= 1, Z2
360°)
+
i sin (180°
+
k. 360°)].
= cos 45°
= cos 135°
+
+
!V2 + !V2i.
-!V2 + !V2i.
i sin 225° = -!V2 - !V2i.
i sin
45°
=
i sin 135° =
= cos 225° +
= cos 315° + i sin 315° =
!V2 - !V2i.
three roots of the cubic equation
xa - 1 + i = O. The three
roots of Example 2 are the solutions of the equation x3 - 1 = O.
1 = l[cos (0° + k. 360°) + i sin (0° + k . 360°)]
= cos (k . 360°) + i sin (k . 360°).
Then
+ k.
Finding the nth roots of a complex number, a + bi, is equivalent to solving the equation xn - (a + bi) = O. Therefore,
DeMoivre's theorem gives a means of solving the general binomial
equation.
For example, the three roots of Example 1 are the
Z3
l-i
= l[cos (180°
These results, if plotted, would lie at the vertices of a square.
ZIX
FIG. 113.
181
..y -1 = cos (45° + k . 90°) + i sin (45° + k . 90°).
Giving k in succession the values 0, 1, 2, 3, the four fourth
roots of -1 are as follows:
1.084i.
x
SERIES
The triangle in each case is inscribed in a circle of radius equal
to the common modulus of the roots.
Example 3.-Find
all the fourth roots of - 1 in rectangular
form.
Solution.-The
modulus of -1 is 1 and its amplitude is 180°.
= -0.7937 - 0.7937i.
= 1.084 - 0.2905i.
Z2
DEMOIVRE'S THEOREM,
NUMBERS,
i sin 240°.
As already pointed out, the n distinct nth roots of a complex
m~
lie at the yertiees of a regular n gon whose center io at
the origin, and whose vertices lie on a circle whose radius is r,
the common modulus of the roots. This is immediately apparent from the general form of the nth root. Whence it is seen
that all the nth roots have the same modulus and hence all lie
at the same distance from the origin. Also their amplitudes
differ by the constant angle 27r, as k is given in s~ccession the
n
values 0, 1, 2, . . . n - 1. Therefore, the points representing
the roots are equally spaced around a circle.
EXERCISES
Find all the roots in Exercises 1 to 14 and express in polar form.
1. x' = 1 - i.
2. x3 = 1 - i.
3. x3 = -1 - i.
4. x3 = h/3 + !i.
6. X3 = !v'3 - !i.
6. x5 = i.
form,
= -! + !V3i, Za = -! - !V3i.
The plotting is shown in Fig. 114.
Note that in Examples 1 and 2 each of the cube roots lies at a
vertex of an equilateral triangle whose center is at the origin.
~
I
l
182
PLANE AND SPHERICAL TRIGONOMETRY
7. x' = -1.
8. x2 = cas 20°
9.
10.
11.
12.
13.
x2 =
x3 =
x3 =
x' =
X4 =
+
COMPLEX
cos a
22.
x' = cas 120 +
+
i sin 120.
-
1 = O.
64 = O.
+ 1 = O.
-
-
0) (a
..
COSn-4
-
30)
+
( )
cos n-4 0
Sin 0
0
4
( )
zero, cos 0 ~
1,
a2
2!
+
a4
4!
a6
-
+
6!
...
the coefficients of the imaginary parts of (1),
+
n(n
-
1)(n -2)S~
- 3)(n --4)
Making the substitutions
.
sm a
+
=
a cosn -1
for 8 and n,
Sin 0
a(a -
)-
8( T
~n_~5-B~#_uL"
8) (a -
28)
3!
-3
cos"-e
O(
E>in8
a(a - 8)(a - 28)(a - 30)(a - 48)
Sin 0 5
cos n-5 8(
.
5!
8
)
Then when n becomes infinite
a3
(3)
0 sin4 8
sin2 8
sin nO = n COSn-I0 sin 8 - n(n - 1) (n - 2) COS,,-38 sin3 8
3!
2!
n (n - 1)( n - 2)( n - 3)
COS,,-2 0
20) (a
= 1-
COS a
Equating
cos n8 = cosn 8 - n(n - 1) COSn-28 sin 2 8
+ ...
sina=a--+---+..
3!
a6
a7
6!
7!
.
In (2) and (3), a is in radians.
If we divide (3) by (2), we get
Let a = n8, then 8 =
while nand
a(a
(2)
<
4!
183
1) .
2.
X4
TRIGONOMETRIC SERIES
124. Expansion of sin n9 and cos n9.-By DeMoivre's theorem
and the binomial theorem,
(1) cosn8 + isin n8 = (cos 8+isin})n
()
= cosn + ni cos,,~ll1'-SinB
n(n
1)
1''1(71 l)(n
2)
.
cos" -- IJSIn" () cosn 30 sIn3 8
2!
3!
n(n - 1) (n - 2) (n - 3)
. 4
4
cosn- 8 SIn
+
8
4!
in(n - 1)(n - 2)(n - 3)(n - 4)
.
+
COsn"-< 0 SIn" 0
*
-'"
5!
Equating the real parts,
+
SERIES
T
+ i = O.
31. x' - V -32 = O.
32. x' - V -243 = O.
30.
8 (~0,-
= cosn
8-
Now, as n becomes infinite ~ = 8 approaches
n
sin 8
~
1, and a - 8 ~ a. Therefore,
Solve the following equations, express the roots in rectangular form, and
represent them graphically:
23. x' - 1 = O.
28. x. + 32 = O.
24. x4 - 1 = O.
29. x4 - i = O.
-
THEOREM,
~ (~ - 1)( ~ - 2)( ~ - 3)
COSn-48 sin4 8
+
4!
a(a - 0)
COSn-2
8 sin
- 8 2
= cosn 8 2!
8
Find all the roots in Exercises 15 to 22 in rectangular form and represent
them graphically.
15. x' = -1.
19. x4 = -1.
16. x' = -S.
20. x6
= -i.
17. x' = i.
21. x.
= 32 (cas 150° + i sin 150°).
18. x' = -i.
DEMOIVRE'S
~
i sin 20°.
cas 140° + i sin 140°.
cas 105° + i sin 105°.
cas 300° + i sin 300°.
-32.
S + SV3i.
14. x' = -S - SV3i.
25. x.
26. x6
27. x'
NUMBERS,
and n = ~, where a is to be held constant
~
8 are to vary. Substituting these values,
. a
sIn
tan a =
* The symbol n!, or~, is used to denote the product 1 . 2 . 3 . . . n, and
is read" factorial n."
COS
.&.
a
a
=
1
-
a3
a5
2
a4
3 .! + 5!
~!
-
+ 4! -
a7
7!
a6
6!
=a+3
a3
2a5
+15+""
)
3
l
184
PLANE AND SPHERICAL TRIGONOMETRY
COMPLEX
125. Computation of trigonometric functions.-Formulas
and (3) may be used to compute the functions of angles.
(2)
Thus,
(3)
. 1
1
(n'1l") 3
Slll-::-:::7r=-::-:::7r--+--'"
18
18
3!
(-(-g-'II")6
5!
=a-
3 log '11"= 1.49145
colog 183 = 6.23419 - 10
colog 3! = 9.22185 - 10
a = 0.17453
sin 6
(5)
log b = 6.94749 - 10
b = 0.000886
=
(
1 - 2!
+ 4! - 61 + . ..
(2i sin (j)n = (eiO
=
)+ ( ~
(j
3!
+
5!
+...
(j7
-
7!
+ . ..
.
e
=
EXERCISES
values prove the following
1.
3,
1
+
('ot2
Ii
=
('.('2
identities:
A,
=
eio - e-io.
Expanding by the binomial theorem,
).
+
-
(eiO)n
'4 (j 4
(j5
.
eiO+ (riO
2
2i sin (j
+ x + 2! + 3! + 4! + . . . .
Now if i(j is substituted for x, where i = V -1,
'2(j
eio
= 1 + i(j + ~2! + ~3! + ~4! + . . .
(j3
e-iO
127. Series for sinn 6 and cosn 0 in terms of sines or cosines
of multiples of 6.-From
(4) of Art. 126,
= 1
,
(j.
4.
2. 1 I tan. @ scc' 8,
2 6;1<e ,-,0" e.
"in 2e
6. cos 20 = cos2 (I - sin2 0 = 2 cos2 8 - 1 = 1 - 2 sin2 o.
6. cos 38 = 4 COS3 8 - 3 cos 8,
126. Exponential
values of sin 6, cos 6, and tan 6.- In algebra
it is proved that if e is the base of the natural system of logarithms, then
x2
x3
x4
(j6
-
eiO
+
1. sin2 (I + COS2 0
Compute the following functions correct to the fourth decimal place and
compare with the tables:
1. sin 20°,
3. tan 30°.
2. cos 25°.
4. sin 45°.
(j4
i sin
Note.-The expressions for sin 0, cos 0, and tan 0, given in (4), (5), and
(6), are called exponential values of these functions.
They are also called
Euler's Equations after Euler their discoverer.
Euler (1707-1783) was one
of the greatest of the physicists, astronomers,
and mathematicians
of the
eighteenth century.
EXERCISES
(j2
-
eiO
.
tan 6 = i(eiO- e-iO
iO)
By means of the exponential
'3 (j 3
=
cos 6 =
(6)
log c = 4.13022 - 10
c = 0.000001349
sin 10° = a - b = 0.17453 - 0.000886 = 0.17364.
From the table of natural functions, sin 10° = 0.17365.
By means of (2), cos 10° may be computed.
2
(j
= cos
Dividing (4) by (5),
colog 185 = 3.72365 - 10
colog 5! = 7.92082 - 10
eX
(j.
i sin
Adding (2) and (3),
5 log '11"= 2.48575
(1)
+
(3) from (2),
(4)
where a, b, c, . . . may be computed as follows:
log '11" = 0.49715
log 18 = 1.25527
log a = 9.24188 - 10
e-iO
Subtracting
b+c-
(j
cos
185
x = -i(j in (1) and reducing as before, we have
Substituting
Then
DEMOIVRE'S THEOREM, SERIES
. . . eio =
(2)
a = 10° = 1\1r.
let
NUMBERS,
n(n2~
e-iO)n
+ n(eiO)n-l(
-riO)
1)(eiO)2(-riO)n-2
+
+
n(n
1)
2~
neiO(-e-iO)"-1
When n is odd, the number of terms in the
when n is even, the number of terms is odd.
n is odd, the terms can be grouped in pairs, the
the second with the last but one, etc. But,
But, by Art. 124, the expressions in the first and second parentheses are equal to cos (j and sin (j, respectively,
~
(eiO)"-2( -e-iO)2
+
(-riO)n.
series is even, and
Therefore, when
first with the last,
when n is even,
l
-------
186
PLANE AND SPHERICAL TRIGONOMETRY
COMPLEX NUMBERS, DEMOIVRE'S THEOREM, SERIES 187
there will be a certain number of pairs and one extra term, which
is the middle term of the series.
From this series, general formulas can be derived for expressing
sinn (Jas a series of sines or cosines of multiples of (J.
By using (5) of Art. 126, cosn (Jcan be dealt with in a similar
manner.
Here special cases only will be given. From these and other
special cases, however, laws can easily be discovered that will
determine the coefficients, and multiples of the angles.
Example I.-Express
sin5 (Jin sines of multiples of (J.
Since sin
.
sm 5
ei8
(J
=
(J
-
- e-i8
2i
'
1_ ei58
24 [
-
5ei38
- 10e-i8+ 5e-i38- e-i58 .
+ 10ei8
2i
]
Grouping in pairs, the first with the last, the second with the
last but one, etc.,
. (J 1
sm5
= 24
ei58
[
. . . sin5
- e-i58
2i
5
(J = n-(sin 50
Example 2.-Express
ei8 - e-iD
2i
2i
-
5 sin
+ 10
3(J + 10 sin
]
.
(J).
sin6 (Jin cosines of multiples of (J..
Since
.
SIn 6 (J
-
ei38 - e-i38
ei8 - e-i8
2i
'
15ei28 - 20 + 15e-i28 - 6e-i48 + e-i68
.
2
]
Hin (J =
1 ei68 - 6ei48 +
= - -25[
Grouping in pairs,
ei68
e-i68
ei48
+ e-i48 + 15 ei28 + e-i28 - 10 .
- -251 [ +2
- 6
2
2
1
.'. sin6 (J = -i~(cos 6(J - 6 cos 4(J + 15 cos 2(J - 10).
Example 3.-Express
cos3 (Jin cosines of multiples of (J.
.
sm6 (J =
cos (J =
Since
l
cos3 (J = 4[
ei38
ei8
+
2
]
e-i38
ei8
e-i8
l ei38
+ 3 +2 ] .
= 4- [ +2
. '. cos3 (J = l[cos 3(J + 3 cos (JJ.
(J
cos
(J in
=
cosines of multiples of (J
eiD
+ e-iD
'
2
l
ei48
+ 4ei28 + 6 + 4e-i28 +
l
ei48
+
cos' (J = -8
[
.. . cos4
e-i48
2
=- 822
[
e-i48
ei28
+4
(J = Hcos 4(J
]
+
e-i28
]
+3.
+ 4 cos 2(J + 3).
EXERCISES
Prove the following identities:
1. sin' 0 = Hcos 40 - 4 cas 20 + 3).
2. cos7 0 = ';~(cos 70 + 7 cas 50 + 21 cas 30 + 35 cas 0).
3. 128 cosS0 = cas 80 + 8 cas 60 + 28 cas 40 + 56 cas 20
4. 64 sin7 0 = 35 sin 0 - 21 sin 30 + 7 sin 50 - sin 70.
6. sin" 0 + cas" 0 = Hcos 40 - 4 cas 20 + 3).
+ 35.
128. Hyperbolic functions.- In Art. 56, the trigonometric
functions were called circular functions because of their relation
to the arc of a circle. There is another set of functions whose
properties are very similar to the properties of the trigonometric
functions.
Because of their relation to the hyperbola, they are
called hyperbolic functions.
They are defined as follows:
r;-x.
(1) Hyperbolic
sine x (written sinh x)
(2) Hyperbolic
cosine x (written cosh x) =
(3) Hyperbolic
tangent x (written tanh x) = ez
(4) Hyperbolic
cotangent x (written coth x) = eX - e-
= ~.
eX + e-X
.
2
eX
cosecant x (written csch x) =
- e-X
+ e-x'
ex + e-X
.
:
(5) Hy p erbolic secant x (written sech x) =
eX
'
+ 3ei8 + 3e-i8 + e-i38
cos4
Since
(6) Hyperbolic
e-i8
2
Example 4.-Express
eX
e-X
2
.
- e-Z
In these formulas e is the base of the N apierian system of
logarithms, and so stands for the number 2.7182818 . . . .
From the definitions, the following relations are evident:
tanh x =
sinh x
cosh x'
1
tanh x = -,
coth x
sech x
=
1 .
cosh x
~
COMPLEX
PLANE AND SPHERICAL TRIGONOMETRY
188
129. Relations between the hyperbolic functions.-8quaring
(1) and (2) and subtracting the second from the first,
cosh' x
-
eZz + 2
sinh' x =
+ e-Zz
e'X
2 + e-2x
=1.
4
-
4
.
. . cosh2 X - sinh2 x = 1.
Dividing
Also,
+ y) =
131. Expression of sinh x and cosh x in a series.
tion.-By
definition and by (1) of Art. 126,
2
pX
+ eXe-1i
e-X
e-(x+II)].
+
.
.'.
Sinh x
Also,
) - (1-x+2!-3r+
X
. ..
= x
+
1
1
sinh (-x) = - sinh x.
cosh (-x) = cosh x.
sinh (x - y) = sinh x cosh y - cosh x sinh y.
cosh (x + y) = cosh x cosh y - sinh x sinh y.
cosh (x - y) = cosh x cosh y + sinhx sinhy.
tanh x + tanh y .
8. tanh (x + y ) =
1 + tanh x tanh y
9. sinh 2x = 2 sinh x cosh x.
x.
130. Relations between the trigonometric
and hyperbolic
functions.-If
in (4) of Art. 126 we substitute iO for 0,
i sin iO = tlei(i6) - e-i<i6)J= -t[e6 - e-6J = - sinh O.
(1)
.'. sin i6 = i sinh 6.
iO for 0 in (5) of Art. 126,
cas iO = t[ei(i6) + e-i(i6)J = t[e6 + e-6] = cosh O.
. '. cos i6 = cosh 6.
x3
3!
x2
2X2
2x4
=22+21+41+'"
[
x3
x3
+
] = x
+
cosh x = t[e'"
=- 2 [( l+x+-+-+'"
2!
189
Computa-
3!
x5
+ 51+
)]
x3
. .
3! +
)]
. . .
x6
. . . .
5! +
e-"'J
+
) + (1-x+
3!
. .
x3
x2
] =1+21+4!+""
X'
2!
x4
XZ
x6
. . . .
. .. cosh x = 1 + 21+ x'
4! + 6! +
Series (1) and (2) for sinh x and cosh x are convergent for all
real values of x. Therefore, for any real value of x the hyperbolic
fuudiu116
3.
4.
6.
6.
7.
(2)
2x5
+ 2x3
3T + 51 +
(1)
cosh x sinh y.
Prove the following identities:
1. sech' x + tanh' x = L
2. coth2 X - csch' x = 1.
Substituting
2x
(2)
EXERCISES
+ sinh'
[
2
x3
eY
Comparing this with the first,
sinh (x + y) = sinh x cosh y
10. cosh 2x = cosh' x
[( 1+x+2!+3!+'"
1
=
-
x2
=2
. --- - e-Y
cosh x sinh y = - +
2
2
= t[ex+y + e-"'eY - e"'e-Y - e-(x+II)].
Adding the last two,
sinh x cosh y + cosh x sinh y = tle"'+11- e-(x+II)J.
And
SERIES
sinh x = -He'"- e-xJ
ex+y - e-(x+II)
.
t[ex+1I - e-XeY
THEOREM,
tan i6 = i tanh 6.
1
e-V
sinh x cosh y = e'" - e-X . eY +
2
2
=
DEMOIVRE'S
(1) by (2),
(3)
By analogy, from (1) we may write
sinh (x
NUMBERS,
ui ;I, I;all
LI; I;U11l1JULeJ.
131'. Forces and velocities represented as complex numbers.Since forces and velocities, to be completely defined, must be
known in magnitude and direction, they are vectors and may be
expressed in the complex number notation.
In Fig. A, OP
represents to scale a force of F lb., making an angle 0 with the
x-axis. By Art. 113
OP = F(cos 0
+ i sin
0).
[lJ
Since OP locates the point P with polar coordinates (F, 0), it
suggests the following notation: Force (F, 0) represents a force
of magnitude or modulus F lb., with direction or amplitude O.
Force (5, 0°) defines point A in Fig. B, and is OA. Force
(5, 135°) locates point P and is OR.
Example I.-Locate
the following:
Force (10, 90°) ; Force ( 8, 180°);
Force (10, 240°); Force (15, 300°).
l
190
PLANE
COMPLEX
AND SPHERICAL TRIGONOMETRY
Similarly, velocity (20 miles, 45°) means that a body is moving
20 miles per hour in a northeasterly direction, as the x-axis will
be taken as the East and West line.
Equation (1) may be written
OP = F cos 8 + iF sin 8 = x + yi
.'
.,.{80.
"..
x
A
x
FIG. B.
FIG. A.
algebraically.
Their sum is a complex number with the x
component equal to the sum of the x's of the forces, and the y
'\;VUlllvllent equal to Lilt;lJUUlvf Lite Ii'8.
Example 2.-Find
the resultant of the forces in Example 1.
Force (10, 90°) = 0 + 10i.
Force ( 8, 180°) = - 8 + Oi.
Force (10, 240°) = -5 - 8.66i.
Force (15, 300°) = 7.5 - 13i.
Force (F, 8) = (0 - 8 - 5 + 7.5)
= -5.5 - 11.66i
+
Then
F = y(-5.5)2
and
= 12.8
8 = tan-1 (-!~:~6)
+ (10 + 0 -
8.66
-
13)i.
(-11.66)2,
= 180° + 64° 44.8' = 244° 44.8'
Force (12.8,244° 44.8') is the resultant. Notice that the angle
8 is in the third quadrant since both components are negative.
Example 3.-The current in a river flowing due south causes a
boat to drift three miles per hour. At the same time a wind from
DEMOIVRE'S
THEOREM,
SERIES
191
the southwest causes the boat to drift to the northeast at the rate
of six miles per hour. Find the resultant velocity of the boat
both in magnitude and direction.
Solution.Velocity
Velocity
where x and yare the rectangular components of OP.
If the problem is to find the resultant or sum of several concurrent forces, first express each of the forces in the rectangular
form. Then by Art. 115, these complex numbers may be added
y
,Y
p
NUMBERS,
(3 miles, 270°)
(6 miles,
45°)
=
=
0 - 3i.
4.24 + 4.24i.
Velocity (V, 8)
=
4.24
+
1.24i.
V = y(4.24)2 + (1.24)2
= 4.32 miles per hour
1.24
8 - t an -1
4.24.
= 16° 18.1'.
Therefore, the boat moves with a velocity of 4.32 miles per hour
in a direction E 16° 18.1' N.
If a set of concurrent forces are in equilibrium, their resultant
must equal zero; that is, x + yi = O. But this can be true when,
and only when, x = 0 and y = O. The sum of all the x-components of the forces will equal zero. Similarly, the summation
of y-components equals zero. This leads to two equations which
can be solved simultaneously.
For other problems the student is asked to solve Nos. 1, 2,3,4,
5, 6, 9, 11, 12 of Art. 69, by the theory of complex numbers.
37, 38, 39, 10 of Cenoral K.:elei"c" d.t the euJ uf
"ili'O .~~
Chapter IX.
ANSWERS TO EXERCISES
Page 170
1. 9 - 5i.
2. - 2 - 3i.
3. 10 + 3i.
10. i(2
+
11. i( -2y2
12.
13.
14.
15.
16.
5.-7 + 17i.
6. 11 - 3i.
9i).
+
5i).
/3 (6 - V2) -1.eV3 + 2Y6)i.
H2y2 - 3Y3) + H2V3 + 3y2)i.
V2 - V3i
17. 1 - iy5i.
-7 - 6i.
18. ! + !i.
19. -2i.
!(3 - i).
23. 1.
7. H7 - 6i).
8. ne23 - 14i).
9. i\,e3 - i).
4.8-9z.
25. a +
24. -1.
26. 2a2 - 1 + 2ai~.
27. a2 - yl - a4 + ai(YI
20. -2 + 2i.
21. -7 - 24i.
22. -3V3 - 7V2i.
-
a" + yl + a2).
b
+ 2Yab
a- b
28. O.
'
29. 1.
~
\.
192
PLANE
AND SPHERICAL
30.i.
32.
TRIGONOMETRY
31.
V3a3
9
35 . 42 .
- 30
-a + (V3a- 3°a3
v'2a3 .
- -r~'
)
- 3 + (6V2
a3
~6
)~..
34 .
-
17
+ 12i.
Page 174
6.
8.
10.
12.
14.
16.
18.
20.
2V3(cos 300° + i sin 300°).
2i(cos 135° + i sin 135°).
2V2(cos 60° + i sin 60°).
2V7(cos 150° + i sin 150°).
4(cos 90° + i sin 90°).
6(cos 270° + i sin 270°).
cos 270° + i sin 270°.
cos 180° + i sin 180°.
22.
24.
v'3(cos
90° + i sin 90°).
3(cos 270° + i sin 270°).
26. 2V5(cos 333° 26'
34. 3V2[cos
37.
3[cos
38. !V3
43. -2V3
47. -y'3
(45°
(90°
+
CHAPTER
SPHERICAL
+ i sin 333° 26').
+ k . 360°) + i
k . 360°)
+
sin (45°
i sin (90°
+
+
k . 360°)].
k .360°)].
+ Ii.
- 2i.
+ i.
Page 181
1. {/2(cos 78° 45' + i sin 78° 45').
\'./2(COR 1OR°1.5'
2(cos 258° 45'
+i
+i
sin 16Ro -15').
sin 2580 45').
-\Y2(cos 348° 45' + i sin 348° 45').
4. cos 10° + i sin 10°;
11. cos 100° + i sin 100°;
cos 130° + i sin 130°;
cos 220° + i sin 220°;
cos 250° + i sin 250°.
cos 340° + i sin 340°.
16. 1 + v'3i;
1 - V3i;
-2.
30.
0.3827 + 0.9239i.
20.
0.7071 + 0.7071i.
-0.2588 + 0.9659i.
-0.9239 + 0.3827i.
-0.3827 - 0.9239i.
-0.9659 + 0.2588i.
0.9239 - 0.3827i.
-0.7071 - 0.7071i.
0.2588 - 0.9659i.
0.9659 - 0.2588i.
XII
TRIGONOMETRY
132. Spherical trigonometry investigates the relations that
exist between the parts of a spherical triangle.
For convenience, a few of the definitions and theorems of
spherical geometry are stated here.
The section of the surface of a sphere made by a plane is a
great circle if the plane passes through the center of the sphere,
and a small circle if the plane does not pass through the center
of the sphere.
The diameter of a sphere perpendicular to the plane of a circle
of the sphere is called the axis of that circle. The points where
the axis of a circle of a sphere intersects the surface of the sphere
are called the poles of the circle.
133. Spherical triangle.-A
spherical triangle is the figure
on the surface of a sphere bounded by three arcs of great circles.
The three arcs are the sides of the trianglc, and the angles formed
by the arcs at the pomts where they
c
meet are the angles of the triangle.
The angle between two intersecting arcs is measured by the angle
between the tangents drawn to the
arcs at the point of intersection.
E
0
If a trihedral angle is placed with
its vertex at the center of a sphere,
the face planes intersect the surface
D
.
. es
FIG. 115.
0 f th e sp h ere m arcs 0 f grea t C1rc
I
which form a spherical
triangle.
The sides of the spherical
triangle
measure
the face angles of the trihedral
angle, and
the angles of the triangle are equal to the dihedral angles of
the trihedral angle.
In Fig. 115, 0 is the center of a sphere.
O-ABC is a trihedral
angle.
AB, BC, and CA are arcs of great circles.
ABC is a
spherical triangle.
Arcs a, b, and c are the measures ofa,{3,
and "y
193
~
194
PLANE
AND
SPHERICAL
SPHERICAL TRIGONOMETRY
TRIGONOMETRY
LBCA and D-OC-E are measured by the same
respectively.
plane angle, as also are LABC and E-OD-C, and LCAB and
C-OE-D.
The sum of the sides of a spherical triangle is less than 360°.
The sum of the angles of a spherical triangle is greater than 180°
and less than 540°.
It is evident that
can be greater than
agreed to consider
sides and angles are
the sides and angles of a spherical triangle
180°; however, to simplify the subject, it is
only those spherical triangles in which the
each less than 180°.
195
formed into another by replacing each side, or angle, by the supplement of its opposite angle, or side, in the polar triangle.
Since the side of a spherical triangle and the corresponding
face angle of
the trihedral angle have the same numerical measure, the plane trigonometric
functions may be taken of the arcs as well as of the plane angles.
Hence
the identities of plane trigonometry
are true for the sides of a spherical
triangle.
A right spherical triangle is one which has an angle equal
to 90°. A birectangular triangle is one which has two right
A
A'
Right
B'
Birectangu!ar
FIG.
angles.
angles.
C"
A trirectangular
a
134. Polar trlangles.- If the vertices ofa spherical triangle
are used as poles and great circles drawn, another triangle is
furmed-mrltcd- the- polar triangle of -tfie-fust;
.
Thus in Fig. 116, A is the pole of a', B the pole of b', C the pole
of c', and A'B'C' is the polar triangle of ABC.
It is evident that, in general, the great circles drawn as stated will intersect
so as to form eight triangles. The one of these is the polar triangle in which
A and A', Band B', C and C'lie on the same side of a', b', and c' respectively.
If one triangle is the polar of another, then the latter is the polar
triangle of the former.
The sides and the angles of a spherical triangle are the supplements, respectively, of the angles and the sides opposite in the polar
triangle, and, conversely.
= 7r
= 7r
- a, B'
- A, b'
triangle is one which has three right
RIGHT SPHERICAL TRIANGLES
FIG. 116.
Thus in Fig. 116, A'
a'
Trirectangular
117.
= 7r
= 7r
- b, C'
- B, c'
These relations are of great importance, for,
theorem be proved with respect to the sides and
spherical triangle, it can at once be applied to the
Thus, any theorem of a spherical triangle may be
= 7r
- c,
= 7r
- C.
if any general
angles of any
polar triangle.
at once trans-
135. In a spherical
three angles, besides
known.
triangle there are six parts, three sides and
the radius of the sphere which is supposed
In general, 7f three of these parts are given the other parts
can fre fmmd;
-
Iftlm-
t.rictngle
-h><t-right
ISphcricd.I -1.1j,wglc,
-tWtt
given parts in addition to the right angle are sufficient to solve
the triangle.
Since there are three unknowns to be found in solving a right
triangle, it is necessary to have any two given parts combined
with the remaining three in three independent relations or formulas. Now, since the five parts taken
B
three at a time form ten combinations,
ten
formulas are necessary and suffi-
cient to solve all right spherical
triangles.
If a, b, c, a, and {3are the five parts 0
',IC
of the triangle, omitting the right
angle, then the ten combinations of
A
these taken three at a time are abc,
FIG. 118.
aba, ab{3, aca, ac{3, aa{3, bca, bc{3,ba{3,
and ca{3. It is necessary to derive a formula connecting the parts
in each of these combinations.
~
PLANE AND SPHERICAL TRIGONOMETRY
196
SPHERICAL TRIGONOMETRY
136. Derivation of formulas for the solution of right spherical
triangles.-Let
0 be the center of a sphere of unit radius, and
ABC a right spherical triangle, with 'Y the right angle, formed by
the intersection of the three planes AOB, AOC, and COB with the
surface of the sphere. Pass the plane BF E through B perpendicular to OA. Then the plane angle BFE measures a, and
a, b, and c have, respectively, the same measures as angles
COB, AOC, and AOB. Further, EB = sin a, FB = sin c,
OF = cos c, and OE = cos a.
FE = EB cot a
FE = F B cos a
FE = OE sin b
FE = OF tan b
From (a) and (b), sin a cot a
.
Sln a
sin a cot a.
sin c cos a.
cos a sin b.
cos c tan b.
sin c cos a.
.
COSa
or
= Sln c -,
cot a
(1)
sin a = sin a sin c.
By analogy, interchanging a and b, a and {:J,
Then
(2)
=
=
=
=
=
sin b = sin {:Jsin c.
(a)
(b)
(c)
(d)
(3)
(4) By analogy,
cot a, or
=~
sin b = tan a cot a.
From (a) and (d), sin a cot a = COSc tan b.
a cot a
tan b cot {:Jcot a or
.. . cos c = sin tan
=
tan b
'
b
(5)
cos c = cot a cot {:J.
(b) and (c), sin c cos a = cos
cos a sin b
cos
.
=
. '. COS a =
Sln c
(6)
cos a = sin
(7) By analogy,
cos {:J = sin
From
From
(b) and (d), sin c cos a
.'.
(8)
(9) By analogy,
=
COS c tan
b
=
COS a = tan b cot c.
cos {j = tan a cot c.
COS
a
(aba)
(ab(3)
(ca(3)
( aa(3)
(baS)
b.
cos c
-;-- tan b, or
Sln c
(J
A
a sin b.
a sin c sin {:J or
.
'
Sln c
{:Jcos a.
a cos b.
137. Napier's rules of circular parts.-The
preceding ten
formulas for the solution of right spherical triangles are included
in a theorem first stated and proved by Napier.
The theorem is
usually stated as two rules known as "Napier's rules of circular
parts."
In the right spherical triangle ABC, omit the right angle at
C and consider the sides a and b, and the complements of a, (3,
and c. Call these the circular parts of the triangle and designate them as a, b, co-a, co-(3, and co-c.
In the triangle or the circular scheme shown in the figure,
anyone of these five parts may be selected and called the mid-
(bc(3)
sin b
sin a = tan b cot {:J.
(abc)
(aca)
From (a) and (c), sin a cot a = cos a sin b.
. '.
From (c) and (d), cos a sin b = cos c tan b.
b
.'. cos c = costana sin
or
b '
(10)
cos c = cos a cos b.
197
(bca)
(ac(3)
FIG. 119.
dIe part; then the two parts next to it are called adjacent parts,
and the other two parts the opposite parts. For example, if
b is chosen as the middle part, then CO-aand a are the adjacent
parts and co-c and co-(3 are the opposite parts.
Napier's rules
are then stated as follows:
(1) The sine of a middle part equals the product of the tangents
of the adjacent parts.
(2) The sine of a middle part equals the product of the cosines
of the opposite parts.
It may assist in remembering the rules to notice the repetition
of a in (1) and of 0 in (2).
Napier's rules may be verified by showing that they give the
ten formulas of Art. 136. A demonstration of the theorem as
given by Napier may be found in Todhunter and Leathem's
"Spherical Trigonometry."
l
198
SPHERICAL TRIGONOMETRY
PLANE AND SPHERICAL TRIGONOMETRY
Example.-Use
applying Napier's rules. Or the formulas can be chosen from
Art. 136.
The quadrant in which the unknown part belongs is determined
by the rules of species.
The work may be checked by applying Napier's rules to the
three parts obtained by the solution of the triangle.
Example I.-Given
a = 30° 51.2', (3 = 71° 36'; find a, b, and
c, using Napier's rules.
Co-a as the middle part and apply rule (1).
sin (co-a)
. '. COSa
= tan
= tan
b tan (co-c).
b cot c, which is formula (8).
Exercise.-Verify
all the ten formulas by Napier's rules.
Napier's rules thus furnish a very convenient way for the
determination of the formulas for the solution of right spherical
triangles.
138. Species.-Two
angular quantities are said to be of the
same species when they are both in the same quadrant, and of
different species when they are in different quadrants.
Since any or all the parts of a right spherical triangle may be
less than or greater than 90°, it is necessary to have a method for
the determination
of the species of the parts. The following
rules will be found to cover all cases:
(1) An oblique angle and its opposite side are always of the same
species.
(2) If the hypotenuse is less than 90°, the two oblique angles and
therefore the two sides of the triangle are of the same species; if the
hypotenuse is greater than 90°, the two oblique angles, and therefore
the two sides, are of opposite species.
These rules are here verified in two cases. As an exercise, the
student is asked to verify them in other cases.
':xanfpleT-ljYlorm1.11a
199
Formulas
To find a, CO-a is the middle
part, a and co-{3 the opposite
parts.
Then sin (co-a) = cos a
=
cos (co-{3), or cos a
. '.
cos a
Construction
cos a sin {3.
a
= COS
;--'
sm {3
b
FIG. 120.
To find b, co-{3 is the middle
part, CO-aand b the opposite parts.
cos b, or cos {3 = sin a cos b.
. . . cos b
To find c, co-c is the middle
Then sin (co-{3)
=
cos (co-a)
cos {3
=
sin
part
a'
and CO-a and co-{3 are the
~;)) \,.n.H. "'UU),
--
.'. cos c = cot a cot {3.
C < 90°, cos c is +. Then the product cot a cot {3must be +,
and this will be true if cot a and cot {3are both + or both -;
that is, if a and {3are both in the same quadrant.
If c > 90°, cas c is -.
Then the product cot a cot {3must
be -; that is, cot a and cot {3must be opposite in sign, and
therefore a and {3must be in different quadrants.
This verifies
rule (2).
cos {3.
Example 2.-From
formula (7) (Art. 136), sin a = cos b
Since sin a is always +, cos {3and cos b must both be + or both
Therefore, both {3and b are in the same quadrant.
This
verifies rule (1).
139. Solution of right spherical triangles.-As
stated before,
when any two parts other than the right angle are given, the
remaining parts of the right spherical triangle can be found.
The necessary formulas can be obtained by taking each of the
unknown parts in turn with the two known parts, and then
To check, use co-c as the middle part with a and b as the opposite parts.
Then
sin (co-c)
= cos a cos b, or cos c = cos a cos b.
Computation
log cos a = 9.93373
log sin {3 = 9.97721
log cos a = 9.95652
a = 25° 12.8' log cot a = 0.22375
log cot {3 = 9.52200
log cos c = 9.74575
c = 56°9.6'
Note.-The
formulas
Art. 136, by selecting
and
.
(cafJ).
log cos {3 = 9.49920
log sin a = 9.70998
log cos b = 9.78922
b = 52° 0.8'
Check
log cos a = 9.95652
log cos b = 9.78922
log cos c = 9.74574
used in this solution could have been taken from
the formulas for the combinations
(aa{j) , (bafJ) ,
200
Example
2 (ambiguous case).-Given a
=
24°
=
8', a
32°
to the base, the triangle is divided into two symmetrical right
triangles, as ACD and ACB in the figure.
The solution of the isosceles spherical triangle is, therefore,
made to depend upon the solution of two right spherical triangles.
141. Quadrantal
trlangles.- When one side of a spherical
triangle is equal to 90°, the triangle is called a quadrantal triangle.
By taking the polar triangle of a quadrantal triangle a right
triangle is obtained, and this can be solved. The supplements of
the parts of the right triangle will give the corresponding parts of
the quadrantal triangle.
Construction
Formulas
To find (j.
sin (co-a)
.
.'.sm{j
=-.
=
cos a cos (co-{3).
A
COSa
cos a
To find b. sin b = tan a tan (co-a).
. . . sin b = tan a cot a.
To find c. sin a = COS(co-a) COS(co-c).
.
sin a
. . . SIn c = --, '
sIn a
EXERCISES
A'
FIG. I2l.
Check.-sin
b
1.
find
2.
find
3.
find
4.
find
= cos (co-c) cos (co-{3),or sin b = sin c sin (3.
Computation
log tan a = 9.65130
log cot a = 0.20140
log sin b = 9.85270
9.96735
b = 45° 25' 40"
68° 3' 36"
111° 56' 24"
b' = 134° 34' 20"
Check
9.61158
log sin c = 9.88536
9.72622
log ~in-/!= 9.96-7~,I)
log sin c = 9.88.736
log sin b = 9.85271
c = 50° 10' 27"
log cos a =
log cos a =
log sin {3 =
{3 =
{3' =
log sin a =
log sin a =
9.92763
9.96028
Since each of the unknown parts is determined from the sine,
there are two values of each unknown part. For this reason it
is called the ambiguous case. The proper
grouping of the parts may be determined
from the rules for species.
By rule (2), when c < 90°, a and {3must
be in the same quadrant. By rule (1),b
C
FIG. 122.
be in the same quadrant.
. '. a, b, c, a, and {j are the parts of one
D
right spherical triangle.
Again by rule (2), when e > 90°, a and {3
are of
opposite
species.
.'. a, b', e', a and {j' are the parts of a right spherical
69° 25' 11", fJ = 63° 25' 3";
50° 0' 0", b = 56° 50' 52", a = 54° 54' 42".
78° 53' 20", a = 83° 56' 40";
77° 21' 40", b = 28° 14' 34", fJ = 28° 49' 54".
61 ° 4' 56", a = 40° 31' 20";
50° 29' 48", fJ = 61 ° 49' 23", a = 47° 55' 35".
70° 23' 42", b = 48° 39' 16";
59° 28' 30", a = 66° 7' 22", fJ = 52° 50' 18".
c = 55° 9' 40", a = 22° 15' 10", b = 51° 53' 0".
Given a = 83° 56' 40", fJ = 151° 10' 3";
a = 77° 21' 50", b = 151° 45' 29", c = 101° 6' 40".
Given tt =%5-° tgt 48", tr =52° &'45"-;---c = 56° 9' 31'''. a = 30° 51' 16" P = 71' at;' 0"
Given a = 100°, b = 98° 20';
c = 88° 33.5', a = 99° 53.8', fJ = 98° 12.5'.
Given a = 92° 8' 23", b = 49° 59' 58";
a = 92° 47' 34", c = 91 ° 47' 55", fJ = 50° 2' 0".
Given fJ = 54° 35' 17", a = 15° 16' 50";
b = 20° 20' 20", c = 25° 14' 38",
= 38° 10' 0".
Given fJ = 83° 56' 40", b = 77° 21' 40"; '"
a = 28° 14' 34", c = 78° 53' 20", a = 28° 49' 54";
151° 10' 3",
a' = 151° 45' 29", c' = 101 ° 6' 40",
"" =
12. Given a = 66° 7' 20", a = 59° 28' 27";
find
b = 48° 39' 16", c = 70° 23' 42", fJ = 52° 50' 20";
b' = 131° 20' 44", c' = 109° 36' 18", fJ' = 127° 9' 40".
13. Solve the isosceles spherical triangle in which the equal sides are
each 34° 45.6', and their included angle 112° 44.6'.
Ans. Equal angles = 38° 59.6'; side = 56° 41'.
14. Solve the isosceles triangle in which the equal angles are each 102°
6.4', and the base 115° 18'.
Ans. Equal sides = 97° 34'; included angle = 116°54.5'.
15. In a quadrantal triangle, c = 90°, a = 116° 44' 48", b
= 44° 26' 21";
find a = 130° 0' 4", fJ = 36° 54' 48", 'Y = 59° 4' 26".
c' = 129° 49' 33"
B
Given c =
a =
Given c =
a =
Given c =
b =
Given c =
a =
5. Given a = 27° 28' 38", fJ = 73° 27' 11";
find
6.
find
7.
rind
8.
find
9.
find
10.
find
11.
find
---------
and {3must
201
By dropping
a perpendicularfrom the vertex of the triangle
10';
find {j,b, and c, using Napier's rules.
--
l
SPHERICAL TRIGONOMETRY
PLANE AND SPHERICAL TRIGONOMETRY
triangle.
140. Isosceles spherical trlangles.- When two sides of a
spherical triangle are equal, it is said to be isosceles.
&:
~
202
PLANE
AND
SPHERICAL
16. In a quadrantal
triangle, c
a = 112° 10' 20", b = 46° 31' 36",
=
TRIGONOMETRY
90°, a
=
121° 20', {3 = 42° 1'; find
edge OA and meeting the faces of the trihedral angle in DE, DF,
and EF. Then LEDF = t is the measure oJ ex.
-2
-2
-2
In the triangle DEF, EF = ED + FD - 2ED. FD cas t.
= 67° 16' 22".
OBLIQUE SPHERICAL TRIANGLES
142. Sine theorem (law of sines).-In
any spherical triangle,
the sines of the angles are proportional to the sines of the opposite
sides.
Proof.-Let
ABC (Fig. 123) be a spherical triangle.
Construct the great circle arc CD, forming the two right spherical
triangles CBD and CAD. Represent the arc CD by h.
By (1) of Art. 136.
'Y
.
sin h
.
and SIn (3 =
sm b
s~n ex
B y division , s~n ex = s~n a, or
sm fJ sm b
sm a =
SIn ex = -:--'
-2
Also, in triangle EOF, EF
Equating
-2
OE + OF
. OF cas a.
these values of EF ,
-
OE2 - ED2 = od and OF2 - FD2 = OD2.
D
s~n fJ.
sm b
A
c
A
FIG. 124.
"'
it may be prO'.'cd that
sin ex sin a
-=-'or-=-'
sin 'Y sin c
sin ex
sin a
sin 'Y
sin c
sinQ_sin~_sin"(
sin a - sin b - sin c'
Formula [44] is useful in solving a spherical triangle when
two angles and a side opposite one of them are given, or when two
sides and an angle opposite one are given.
Note.-While, in the figure, D falls between A and B, the theorem can be
as readily proved if D does not fall between A and B.
143. Cosine theorem (law of cosines).-In
any spherical
triangle, the cosine of any side is equal to the product of the cosines
of the two other sides, increased by the product of the sines of these
sides times the cosine of their included angle.
Proof.-Let
ABC (Fig. 124) be a spherical triangle cut from the
surface of a sphere, with center 0, and radius OA chosen as unity.
At any point D in OA, draw a plane EDF perpendicular to the
c,'Y'
B
FIG. 125.
B
.. .
20E
-2
But OED and OFD'are right triangles, then
sin h
0
[44]
-2
= -2
OE + OF -
-2
-2
20E . OF cas a = ED + FD - 2ED . FD cas t.
- 2 -2
-0
-2
Or 20E . OF cas a = OE - ED + OF"- FD + 2ED . FD cos t.
-:--'
sm a
c
III a. ;:;lmila.l niannel
203
SPHERICAL TRIGONOMETRY
.
a'
FIG. 126.
-:\hking these substitutions, rlivirling by the coefficient of cos a,
and arranging the factors, there results
OD . OD
ED . FD cos t.
cos a =
ED OF + OE OF
[451]
.'. cos a = cos b cos c
Similarly,
+ sin
b sin c cos Q.
[452]
cos b = cos a cos c
+ sin
a sin c cos
[453]
COS C
~.
= cos a cos b + sin a sin b cos "(.
In Fig. 124, both band c are less than 90°, while no restriction
is placed upon exor a. The resulting formulas are true, however,
in general, as may easily be shown.
In Fig. 125, let ABC be a spherical triangle with c > 90° and
b < 90°. Complete the great circle arcs to form the triangle
DCA,
in which AD
=
(180°
-
c)
<
90°.
The parts
of DCA
180° - c, 180° - ex,180° - a, fJ, and b. Then by [451],
cos(180° -a) = cos b cas (180° -c) +sin b sin (180° -c)cos(180°
. . . cas a = cas b cos c + sin b sin c cos ex.
are
-ex).
l
204
PLANE
AND
SPHERICAL
01.'
= -
cos {3' cos "I'
+
[471]
. . . COSa = - cos ~ cos 1
[4711]
[473]
sin {3' sin "I' cos a'.
+
sin ~ sin 1 cos a.
[4611] cos ~ = -cos a cos 1 + sin a
sin 1 cas b.
cas 1 = -cas
+
a cas ~
b
FIG. 127.
sides to find
triangle with given sides a, b, and c.
From [451],
cos a - cos b cos c
(a )
COSa sin b sin c
In order to adapt this formula to logarithmic computation,
proceed as follows:
(1) Subtracting each member of formula (a) from unity,
1
Then
-
cos a
= 1-
2 sin 2 !a =
2
cos (b.- c).- cos a.
1
cos 2~ --
[483]
1
cos 21 --
- b + c).
- a)
sinbsinc'
sin 5 sin (5 - b).
sin a sin c
sin 5 sin (5 - c)
sinasinb'
sin 5 sin (5
----...
Isin (s - a) sin (s - b) sin (s - c) .
Isin (s - b) sin (s - c)
tan!a =
2
'\J sin s sin (s - a) = '\J
sin s sin 2 (s - a)
(sin (s - a) sin (s - b) sin (s - c)
B Y Wfl't' mg r =
sin s
'"
[493]
sm b sm c
- c) sin !(a
1
. 2 201.- sin !(a + b sin
. . . sm
b sin c
sinasinb
[4811]
[4911]
- c-
sin (5 - a) sin (5 - c).
sin a sin c
sin (5 - a) sin (5 - b) .
1
sin 2~ =
1
sin 21 --
tan-a
1
2
'
T
= sm
. (5 a) .
-
In a like manner the following are obtained:
sm b sm c
But cos (b - c) - cos a = -2 sin!(b - c + a) sin Hb
Also sin !(b - c - a) = -sin Ha - b + c).
c),
sin (5 - b) sin (5 - c) .
sinbsinc
. 1
.'. sm 2a =
1
cos 2a =
[491]
cos a.- co~ b cos c.
by [28].
-
b + c = 2(s - b).
[481]
sin a
sin ~ cos c.
145. Given the three
-
(2) By adding each member of formula (a) to unity, and carrying out the work in a similar manner to that in (1), the following
are obtained:
Similarly
[463]
a
In like manner the following are obtained:
This formula expresses a relation between the parts of a polar
triangle.
But the relation is true for any triangle, since for every
spherical triangle there is a polar triangle and conversely.
[461]
a + b + c = 2s.
a + b - c = 2(s
N ow let
Then
and
Exercise.-Draw
a spherical triangle in which the two sides b
and c are each greater than 90°, and verify formula [451],
144. Theorem.-The
cosine of any angle of a spherical triangle
is equal to the product of the sines of the two other angles multiplied
by the cosine of their included side, diminished by the product of the
cosines of the two other angles.
Let ABG be the spherical triangle of which A' B'G' is the polar
triangle.
Then a = 180° - 01.',b = 180° - {3', c = 180° - "I',
and a = 180° - a'.
Substituting these values in [451] and simplifying,
cos
205
SPHERICAL TRIGONOMETRY
TRIGONOMETRY
a)
1
T
tan 2~ =
sin (5 - b)'
1
T
.
tan nI =
2
sin C5- c)
146. Given the three angles to find the sides.- If in the
formulas [47] and [48] the parts of the spherical triangle be
replaced by their values in terms of the parts of the polar triangle,
the following formulas are obtained, where S = Ha + (3+ 'Y):
206
PLANE
AND
SPHERICAL
[501]
cos
!b
2
Taking this proportion by composition and division,
cos (S - ~) cos (S - "f).
cos ! a -2
[501]
.
sm
~
(S
~cos
=
[50s]
! -- ~cos
cos 2c
[511]
sin!a
2
=
[511]
sin!b
2
=
[51s]
. 1
Sln-c
=
2
~
-
-:
It
I!
.
sm
(s - b) + sin (s - a)
tan ta + tan t{3
-- sin
sin (s - b) - sin (s - a)'
tan ta - tan t{3
"f
a) C?S (S
-
"f).
smasm"f
(S - a) COS(S - ~).
sin a sin ~
cos
cos ~S - a).
~
sm~sm"f
cos
cos ~S ~
smasm"f
S cos (S
- cos smasm~
.
.
But
"f)
Substituting
I
~
~ cas (8
- cas 8
- a) cas (8 - (3) cas (8
sin tea - b).
tan tea - ~)
= sin tea + b)
cot h
Again, multiplying
1')"
-
~).
"f).
No/e.-Since 90° < S < 270°, cos S is negative.
Also, since, in the polar
triangle, any side is less than the sum of the two others,
(J) + (". - ')'), or (J + ')' - a < "..
(J + ')' - a > -"..
and cos (S - a) is positive.
. . -!... < S - a <
!""
Similarly, it can be shown that cos (S - (J) and cos (S - ')') are each
positive.
This makes the radical expressions of this article real.
Further, the positive sign must be given to the radicals in each case, for
!a, !b, and !c are each less than 90°.
".
-
147. Napier's
a
<
(".
-
analogies.-Dividing
[491] by [491],
sin (s - b)
- sin (s - a)
tan !{3
tan!a
[491J{Y [491],
sin (s - e)
tan !a tan t{3
sins'
1
Taking
this proportion
+
by composition
and division,
+ Rin
(8 - c)
1
- sin s - sines -c}
1 - tan ta tan tf3
are obtained:
tan tb = R cos (S
tan tc = R cos (S
[521]
[52s]
~).
+ ~)
[54]
- cas 8
,
cas (8 - a) cas (8 - (3) cas (8 - 1')
In like manner the following
Further,
sin tea tan tea - b)
= sin tea
tan te
tan ta = R cos (S - a).
[521]
(1), there results
Replacing a, b, e, a, and (3by their values in terms of the parts
of the polar triangle, [53] becomes
1
-cas 8 cas (8 - a)
tan 2a =
'\J cas (8 - (3) cas (8 - 1')
By writing R =
sin tea + (3).
t{3
these values in formula
[53]
.
Dividing [511] by [501]'
= cas (8 - a)
+ tan
tan ta
(1)
tan ta - tan t{3 = sin tea - (3)
2sint(2s - a - b) cost(a - b)
sines - b) + sines - a)
AI so
- 2 cost(2s - a - b) sint(a - b)
sin (s - b) - sin (s - a)
tan te
tan tea - b)"
~).
-
207
TRIGONOMETRY
SPHERICAL
TRIGONOMETRY
t~n !a t~n !/3
Rin 8
(2)
But
+ tan ta tan t{3
1 - tan ta tan tf3 sin s + sin (s Also
sin s - sin (s 1
cas ta cas t{3 +
casta cas t{3 2sint(2s
e)
e) = 2 cas t(2s
sin ta sin !(3 cas tea - (3)
sin!a sin t{3 - cas tea + (3)'
tant(a + b).
- e) coste
- e) sin te =
tan te
Substituting these values in formula (2), there results
cos tea - ~).
tan tea + b)
[55]
= cos tea + ~)
tan!c
Replacing a, b, e, a, and (3 by their values in terms of the
parts of the polar triangle, [55] becomes
[56]
Equations
analogies.
+ ~)
tan tea
cot h
cos !(a
= cos t( a
-
b).
+ b)
[53], [54], [55], and [56] are known
as Napier's
~
208
PLANE
AND
SPHERICAL
SPHERICAL
TRIGONOMETRY
By making the proper changes in a, b, e, a, {3,and 1', the corresponding formulas may be written for the other parts of the
triangle.
148. Gauss's equations.-Taking
the values of sin ta, cos ta,
sin t{3, and cos t{3, and combining the functions of !a with those
of t{3, there result the four following forms:
..!
sm
fsin (s - b) sin (s - e)
.!{3--
2 a COS 2
=
'\J
-
sin (s
sin s sin (s - b)
sin b sin e
sin a sin e
b) ./sin s sin (s - e)
sin a sin b
=
sin (s - b)
cos
sin e
.!21'.
(1)
.
- ~ Isin s sin (s - a) sin (s - a) sin (s - e)
cos .!
2a sm .!{3
2
-''\j
sin b sin e
sin a sin e
sin (s - a) fsinssin (s - e) sin (s - a)
=
= sin e cos .!21'. (2)
sin e
'\J sin a sin b
cosJacos.!{3
= sin s sin (s - a) sin s sin (s - b)
2
2
sin b sin e
sin a sin e
..! 21'. (3)
-- sin s sin (8 - a) sin (s - b) - sin s sm
sin e
sin a sin b
sin e
fsin (s - b) sin (s - e) JSin (s - a) sin (s - e)
'"
sin b sin e
sin a sin e
sin (s - e) ,,' 1
e) sin (s - a) sin (s - b)
dn 21'. (4)
sin a sin b
.
.
sm .!2 a sm .!{32
=
sin (s
-
sin e
sin e
Adding (1) and (2), there results
sin
1
(s - b) +
tea + (3)= co.s21'[sin
Sin e
sin (s
1
. c~s 21' 1 2 sin t[2s
= 2 sm
'Ie cas 2e
-
-
a)]
(a
+ b)] cas
e)]
(2 S
-
.
=
But cas te2s
[68]
sm 1
. 1 2"1' 1 2
2 sin 2e cos 2e
- e) = cas tea +
.' . cos tc cas tea
COS
1
2
+ ~) =
sin
cos .!2
. 1 I' 1 2 cos ![2s
= 2 sm
2"e cos 2e
[69]
h cas tea +
b).
-
+
(a
b)] sin tea
- b).
.'. sin tc sin tea - ~) = cos h sin tea - b).
Adding (3) to (4), there results
. 1
COStea - {3) = Si~ 21'[sin s
sm e
. 1
sm 2"1'
= 2 sm.
[60]
+ sin
(s
-
e)]
.
1
1
2"e
cas 'le
1 (2
2 sin 2"
s
-
.'. sin tc cos tea - ~) = sin h sin tea
1
e ) cos 2"e.
+
b).
Equations [67], [68], [69], and [60] are' known as Gauss's
equations,
or Delambre's
analogies.
Geometric proofs of
Gauss's equations and Napier's analogies can be found in Todhunter and Leathem's "Spherical Trigonometry."
Exereise.-Derive
[60] from [67] by using the parts of the polar
triangle.
149. Rules for species in oblique spherical triangles.-(l)
If a side (or an angle) differs from 90° by a larger number of degrees
than an()ther side (or angle) in the triangle, it is of the same species
as its opposite angle (o/' side).
Since all angles and sides of a spherical triangle are each less
than 180°, in order to verify this rule, it is necessary to show that
cas a and cos a, for example, have the same sign when
I
From [461],
a-90°
eos a
=
I
a-90°
I > [b -
b sin c cos a.
cos b cos c
sinbsinc
I > Ib -
90° I, a is nearer 0 or 180° than b,
Icos a I > Icos b I, and,
I cos
90° I.
+ sin
cas b cos c
cos a
.'.COSa=
. '.
b).
results
sin tea - {3)= co.s'I1'{sin(8 - b) - sin (s - a)]
sm e
and therefore
unity,
'.1
e ) Sin
2e.
209
1
Since
Subtracting
-
(2) from (1), there
tea - b).
But sin t[2s - (a + b)] = sin te.
[67]
.' . cas tc sin tea + ~) = cos h cas tea - b).
(4) from (3), there results
. 1
COStea + (3) = Si~ 21'[sin s - sin (s
sm e
Subtracting
TRIGONOMETRY
a
I > I cos
since cos e cannot exceed
b cos e
I.
Further, the denominator will always be positive.
Then the sign of cos a is the sign of the numerator of the fraction. That is, cos a has the same sign as cas a; therefore a and a
are in the same quadrant.
~
PLANE AND SPHERICAL
210
TRIGONOMETRY
For example, suppose a = 120°, b = 70°, c = 130°. Since
1120° - 90° I > I 90° - 70° I, a is in the second quadrant with a.
Also, since 1130° - 90° I > I 90° - 70° I, 'Y is in the 'second
quadrant with c. This leaves (3 undetermined in quadrant.
It
is determined by the second rule, which follows.
(2) Half the sum of two sides of a spherical triangle must be of
the same species as half the sum of the two opposite angles.
From
1
]55], tan '2(a + b)
=
1 cos .l(a
2
tan '2c
.
-
-
!c > O. Also, since (a - (3) < 180°,
cos!(a - (3) > O. Therefore, tan!(a + b) and cos!(a + (3)are
of the same sign. But a + b and a + {3must each be less than
therefore,
!(a
+
b) and !(a + (3) must
each
sin a
sin b
sin c
(1)
= sin {3= sin 'Y'
1
r
tan-a = .
2
sm (s - a) '
(s - a) sin (s. - b) sin (s - c). (2)
r = ~sin
sin a
sm s
where
- cas
8
R -"'\JCOS(8 - a) cos (8 - {3)cas (8 - 'Y)
- (3).
(4)
(5)
(6)
tan !(a + {3) cos !(a - b).
= cos!(a+b)
coth
(7)
Given the three sides to find the three angles.
=
Example.-Givena
46° 20' 45", b = 65° 18' 15", c = 90° 31' 46";
to find a, {3,and 'Y.
Construction
Formulas
1
tan '2a
=
B
fJ'
r.
sin (s
-
a)
a
1
tan '2{3=
r.
sin (s - b)
1
r.
tan '2'Y=
sin (s - c)
A
FIG. 128.
r
= "'\JIsin
(s
-
a) sin~ s
-
b) sin (s - c).
sm s
Comp1dation
a
=
46° 20' 45"
log sin (s
b = 65° 18' 15"
log sin (s
c = 90° 31' 46"
2s = 202° 10' 46"
s
s- a
s- b
s - c
2s
=
=
=
=
=
A
-
a) = 9.91200
- b) = 9.76697
log sin (s - c)
colog sin s
101° 5' 23"
54° 44' 38"
35° 47' 8"
10° 33' 37"
202° 10' 46"
check
=
=
log r2 =
log r =
log tan !a =
.
. . !a =
log tan !{3 =
.
. . !{J =
log tan h =
... h =
9.26309
0.00819
8.95025
9.47513
9.56313
20° 5' 15"
9.70816
27° 3' 12"
0.21204
58° 27' 45"
Check by the sine law.
tan ta = R cas (8 - a),
r--
sin!(a
be less
than 180°. Then!(a + b) and!(a + (3)must both be in the first
quadrant or both be in the second quadrant, since tan !(a + b)
and cos !(a + (3)are of the same sign.
150. Cases.-In
the solution of oblique spherical triangles,
the six following cases arise:
CASE I. Given the three sides.
CASE II. Given the three angles.
CASE III. Given two sides and the included angle.
CASE IV. Given two angles and the included side.
CASE V. Given two sides and an angle opposite one of them.
CASE VI. Given two angles and a side (JPPo8'iteone of them.
Any oblique spherical triangle can be solved by the formulas
derived in the previous articles.
In selecting a formula choose
one which includes the parts given and the one to be found.
The following list of formulas, together with the corresponding
formulas for other parts, is sufficient for solving any spherical
triangle:
where
b)
= sin !(a + (J)
tan !c
tan!(a + b) cos!(a - {3).
= cas !(a + {3)
tan !c
tan!(a
- {3) sin!(a - b).
= sin !(a + b)
cot h
151. Case I.
.
-
tan!(a
(3).
Since !c < 90°, tan
360°, and,
211
SPHERICAL TRIGONOMETRY
EXERCISES
(3)
1. Given a = 68° 45', b = 53° 15', C = 46° 30';
find
a = 94° 52' 40", 13= 58° 56' 10", l' = 50° 50' 52".
l
212
2.
find
3.
find
4.
find
5.
find
6.
find
PLANE
Given a
a
Given a
a
Given a
a
Given a
a
Given a
a
=
=
=
=
=
=
=
=
=
=
AND
SPHERICAL
TRIGONOMETRY
SPHERICAL TRIGONOMETRY
70° 14' 20", b = 49° 24' 10", c = 38° 46' 10";
110° 51' 16", fJ = 48° 56' 4", l' = 38° 26' 48".
50° 12.1', b = 116° 44.8', c = 129° 11.7';
59° 4.4', fJ = 94° 23.2', l' = 120° 4.8'.
68° 20.4', b = 52° 18.3', c = 96° 20.7';
56° 16.3', fJ = 45° 4.7', l' = 117° 12.3'.
96° 24' 30", b = 68° 27' 26", c = 87° 31' 37";
97° 53' 0", fJ = 67° 59' 39", l' = 84° 46' 40".
31° 9' 13", b = 84° 18' 28", c = 115° 10' 0";
4° 23' 35", fJ = 8° 28' 20", l' = 172° 17' 56".
!(a
EXERCISES
Given a =
a =
Given a =
a =
Given a =
a =
Given a =
a =
-
'2
b)
(3)
(3)
=
=
=
=
9.55562
9.99638
!c
(3)
=
log tan Ha + b) =
log cos Ha + (3) =
colog cos Ha - (3) =
log tan !c =
. . . !c =
0.88742
82° 37'
0.97657
9.10893
0.01785
0.10335
51° 45.3'
find
a = 100° 49' 30", fJ = 95° 38' 4", 'Y = 65° 33' 10".
2. Given a = 88° 21' 20", b = 124° 7' 17", 'Y = 50° 2' I";
Construction
-
log tan!(a
log sin Ha +
colog sin Ha log tan
. . . Ha +
9.46649
16° 19'
9.93485
9.97360
0.97897
EXERCISES
1. Given b = 99° 40' 48", c = 64° 23' 15", a = 97° 26' 29";
153. Case III. Given two sides and the included angle.The sum and the difference of the two unknown angles can be
found by [54] and [56]. The unknown side can be found by
either [53] or [55] j together, they furnish a check on the work.
Example.-Given
a = 103° 44.7', b = 64° 12.3', 'Y = 98° 33.8';
find a, f3, and c.
Formulas
log tan Ha - (3) =
. . . Ha - (3) =
log cot h =
a = 98° 55.9'
log cos Ha - b) =
f3 = 66° 18'
colog cos Ha + b) =
log tan Ha + (3) =
0.55138
0.10338
. . . !c = 51° 45.3'
129° 5' 28", fJ = 142° 12' 42", l' = 105° 8' 10";
135° 49' 20", b = 146° 37' 15", c = 60° 4' 54".
59° 4' 28", fJ = 94° 23' 12", l' = 120° 4' 52";
50° 12' 4", b = 116° 44' 48", c = 129° 11' 42".
107° 33' 20", fJ = 127° 22' 0", l' = 128° 41' 49";
82° 47' 34", b = 124° 12' 31", c = 125° 41'-43".
102° 14' 12", fJ = 54° 32' 24", l' = 89° 5' 46";
104° 25' 8", b = 53° 49' 25", c = 97° 44', 18".
From [54],
_1 (a - (3) = cot! sin Ha
tan
2'Y - 2
From [56],
cos Ha
1
tan - (a + (3) = cot! 2'Y,
2
From [53],
sin Ha
tan ~c = tan !(a - b)
2
sin Ha
From [55],
cos l (a
1
1
t an,t = tan -(a + b)
2
<)os i (a
Computation
log cot h = 9.93485
a = 103° 44.7'
log sin Ha - b) = 9.52923
b = 64° 12.3'
b) = 83° 58.5' colog sin Ha + b) = 0.00241
!(a - b) = 19° 46.2'
h = 49° 16.9'
152. Case II. Given the three angles to find the three sides.For the solution, use tan !a = R cos (8 - a) and the corresponding forms, and proceed as in Case I.
1.
find
2.
find
3.
find
4.
find
+
213
- b) .
-
find
a = 63° 22' 56", fJ = 132° 13' 58", c = 58° 58' 24".
3. Given b = 156° 12.2', C = 112° 48.6', a = 76° 32.4/;
find
a = 63° 48.7', fJ = 154° 4.1', 'Y = 87° 27.1'.
4. Giycn a = 70° 20' 50", b = 38° 28', 'Y = 52° 29' 45";
find
a = 107°47' 7", fJ = 38° 58' 27", C = 51° 40' 54".
5. Given a = 135° 49' 20", C = 60° 4' 54", fJ = 142° 12' 42";
find
a = 129° 5' 28", l' = 105° 8' 10", b = 146° 37' 15".
154. Case IV. Given two angles and the included side.This case, like the preceding, is to be solved by Napier's analogies,
using the four forms in a similar manner.
EXERCISES
1. Given a = 59° 4' 25", fJ = 88° 12' 24", C = 47° 42' I";
a = 50° 2' I", b = 63° 15' 15", l' = 55° 52' 42".
2. Given a = 63° 45.6', fJ = 95° 56.7', C = 52° 27.8';
find
- b) .
a
+ (3)
-
-'
+ (3).
-
-
'YIC
b
FlG. 129.
find
a = 61° 41.3', b = 77° 29.4', l' = 53° 53.5'.
3. Given a = 125° 41' 44", l' = 82° 47' 35", b = 52° 37' 57";
find
a = 128° 41' 46", C = 107° 33' 20", fJ = 55° 47' 40".
4. Given fJ = 34° 29' 30", l' = 36° 6' 50", a = 85° 59' 0";
find
b = 47° 29' 20",
C
= 50° 6' 20", a = 129° 58' 30".
155. Case V. Given two sides and the angle opposite one
of them.-In
this case the angle opposite the other side can
be found by the sine law, when the other side and angle can be
~
214
PLANE
SPHERICAL TRIGONOMETRY
AND SPHERICAL TRIGONOMETRY
found by Napier's analogies. For example, given a, b, and a,
to find c, {3,and 1', use the following formulas:
sin {3 =
sin
~
sin a.
1
Sln a
1
1
1
. 1
sm 2"(a + (3).
tan 2c = tan -(a - b)
2
sin Ha -
FIG. 130.
+ (3).
cos tea . 1(
a + b).
1(
cos 2"
a
t an 2c = tan -(a
2
+ b)
1
1
cot 21' = tan -(a
2
1
1
- (3) s~n 2"(a -Sln ~2
.
costea + b)
cot 21' = tan -(a + (3)
2
cosHa-b)'
A check is obtained by the agreement in the values of tc and
h from the different formulas.
Since {3 is determined from sin {3, there will be two values
of {3less than 180°, both of which may enter into a triangle.
By
the first rule for species
(Art. 149), if
and {3must be the same species.
Otherwise, both values of {3may
tion of the second rule for species
triangles are possible.
Example.-Given
a = 148° 34'
I90° - b I > I 90°
-
a
I, b
This definitely determines {3.
be admissible.
The applicawill show whether or not two
24", b = 142° 11' 36", and
Cl
= 7° 18' 24".
C2
= 62° 8' 36".
215
1'1 = 6° 17' 36".
1'2 = 130° 21' 12".
This case is the ambiguous case in oblique spherical triangles,
and is analogous to the ambiguous case in plane trigonometry.
In practical applications, some facts about the general shape of
the triangle may be known which will determine the values to be
chosen without having recourse to the rules for species. A complete discussion of the ambiguous case may be found in Todhunter
and Leathem's "Spherical Trigonometry," pages 80 to 85.
EXERCISES
1. Given a = 46° 20' 45", b = 65° 18' 15", IX = 40° 10' 30";
find
2.
find
3.
4.
5.
find
C, = 90° 31' 46", {3, = 54° 6' 19", 'Y1= 116° 55' 26";
C2= 27° 23' 14", {32= 125° 53' 41", 'Y2= 24° 12' 53".
Given a = 99° 40' 48", b = 64° 23' 15", IX = 95° 38' 4";
c = 100° 49' 30", {3= 65° 33' 10", 'Y = 97° 26' 29".
Solve and check, a = 31° 40' 25", b = 32° 30', IX = 88° 20'.
Solve and check, a = 149°, b = 133°, IX = 146°.
Given a = 62° 15' 24", b = 103° 18' 47", IX = 53° 42' 38";
c, = 153° 9' 36", {3, = 62° 24' 25", 'Y1= 155° 43' 11";
C2= 70° 25' 26", {32= 117° 35' 35", 'Y2= 59° 6' 50".
156. Case VI. Given two angles and the side opposite one
of them.-In
this case use the same formulas as in Case V, and
apply the rules for species when there is any question as to the
number of solutions.
EXERCISES
sin b sin a
.
Sln {3 =
.
.
log sin 142° 11' 36" =
log sin 153° 17' 36" =
colog sin 148° 34' 24" =
log sin {3 =
{31=
{32=
sm a
9.78746
9.65265
0.28282
9.72293
31° 53' 42"
148° 6' 18"
Here, since 190° - 142° 11' 36" I < 190° - 148° 34' 24" \, both
values of {3may be admissible.
Since Ha + b) = 145° 23' is in
the second quadrant,
as also are Ha + (31)= 90° 35' 36" and
tea + (32) = 150° 42', they are of the same species by the second
rule for species. Hence, both {31and {32are admissible values to
use.
The student can complete the solution and find the following
values:
1. Given IX =
find
b, =
b2 =
2. Given IX =
find
29° 2' 55", {3 = 45° 44' 6", a = 35° 37' 18"
59° 12' 16", c, = 82° 17' 5", 'Y1= 124° 17' 52";
120° 47' 44", C2= 150° 50' 51", 'Y2= 156° 2' 24".
73° 11' 18", {3= 61° 18' 12", a = 46° 45' 30";
b = 41° 52' 35", C = 41° 35' 4",
'Y
= 60° 42' 47".
3. Solve and check, IX = 122°, {3 = 71°, a = 81°.
4. Solve and check, IX = 37° 42', {3= 47° 20', b = 41° 50'.
5. Given a = 36° 20' 20", {3= 46° 30' 40", a = 42° 15' 20";
find
b, = 55° 25' 2", c, = 81 ° 27' 26", 'Y1= 119° 22' 28";
b2 = 124° 34' 158",C2= 162° 34' 27", 'Y2= 164° 41' 55".
157. Area of a spherical trlangle.-Let
r be the radius of the
sphere on which the triangle is situated, ~ the area of the
triangle, and E the spherical excess.
By spherical geometry, the areas of any two spherical triangles
are to each other as their spherical excesses. Now the area of a
trirectangular
triangle is tn"r2, and its spherical excess is 90°.
Then
~:tn-r2
=
E:90°.
l
216
PLANE
AND SPHERICAL
... A
[61]
=
TRIGONOMETRY
SPHERICAL
6. Given a = 49° 50', {3= 67° 30',
':Cr2E
radius;
180°'
When all the angles of the triangle are known, the spherical
excess and, therefore, the area are easily computed.
If the angles
are not all given, but enough data are known for the solution of
the triangle, the angles may be found by Napier's analogies, and
then the area may be computed by the above formula.
158. L'Huilier's
formula.-This
is a formula for determining the spherical excess directly in terms of the .sides. It may be
derived as follows: Since E = a + {3+ I' - 180°,
sin tea + (3+ I' - 180°)
. lE =
tan
4
cas tea + (3+ I' - 180°)
2 sin tea + (3 + I' - 180°) cas tea + (3 - I' + 180°)
= 2 cas tea + (3
+ I' - 180°) cos tea + (3 - I' + 180°)
sin !(a + (3) - C?s h,
by [29J and [31J.
= cas tea + (3)
+ sm h
cas h cas tea - b).
B y [57 ], sin -!(a + (3) =
2
cas !c
By [58 ], cas
l2 a
e
cas tea + b).
+ (3) = sin h cos
!c
By making these substitutions,
tan ~E = I ::---l(
4
\COS "2 a
=
=
~v
cot
+ + cas "2C) ~I'
-2 sin tea - b + c) sin tea - b - C)
2 cas tea + b + c) cos tea + b - c)
~~
b)
(
I
sin tes - b) sin tes - a)
cas !s cas tes - c)
[62].' . tan tE = ytan!s
)
sin s sin (s
cot 1.
21'
-
c)
'\jsin (s - a) sin (s - b)
.
tan t(s - a) tan t(s - b) tan t(s - c).
EXERCISES
of radius 6 in., a = 87° 20' 45", {3 = 32° 40' 56", and
= 77° 45' 32".
Find the area of the triangle.
Ans. 11.176 sq. in.
2. Given a = 56° 37', b = 108° 14', c = 75° 29'; find E.
Ans. E = 48° 32' 35".
3. Given a = 47° 18', b = 53° 26', C = 63° 54'; find E.
Ans. E = 24° 29' 50".
1. On a sphere
y
4. Given a = 110° 10', b = 33° l' 45",
C
TRIGONOMETRY
= 155° 5' 18"; find E.
Ans. E = 133° 48' 53".
6. Given a = b = C = 60°, on a sphere of 12-in. radius; find the area of
the triangle.
Ans. 79.38 sq. in.
'Y
217
= 74° 40', on a sphere of 10-in.
find the area of the triangle.
Ans. 20.94 sq. in.
7. Given a = 110°, {3= 94°, c = 44°, on a sphere of 10-ft. radius; find
the area of the triangle.
Ans. 128.15 SQ. ft.
8. Given a = 15° 22' 44", C = 44° 27' 40", {3 = 167° 42' 27", on a sphere
of radius 100 ft.; find the area of the triangle.
Ans. 248.32 sq. ft.
9. Find the area of a triangle having sides of 1 ° each on the surface of
the earth.
Ans. 2070 sq. miles.
APPLICATIONS
OF SPHERICAL
TRIGONOMETRY
159. Definitions and notations.-In
all the applications of
spherical trigonometry
to the measurements
of arcs of great
circles on the surface of the earth, and to problems of astronomy,
the earth will be treated as a sphere of radius 3956 miles.
A meridian is a great circle of the earth drawn through the
poles Nand S. The meridian
N
NGS passing through Greenwich, England, is called the
principal meridian.
The longitude of any point P
on the earth's surface is the
angle between the principal
meridian NGS and the meridian W
NPS through P. It is measured by the great circle arc,
CA, of the equator between the
points where the meridians cut
the equator.
S
If a point on the surface of the
FIG. 131.
earth is west of the principal
meridian, its longitude is positive. If east, it is negative. A
point 70° west of the principal meridian is usually designated as
in "longitude 70° W." "Longitude 70° E." means in 70° east
longitude.
The letter 0 is used to designate longitude.
The latitude of a point on the surface of the earth is the
number of degrees it is north or south of the equator, measured
along a meridian.
Latitude is positive when measured north of
the equator, and negative when south. The letter <Pis used to
designate latitude.
160. The terrestrial
triangle.-Given
two points on the
earth's surface, H with latitude <PIand longitude Ol, and P with
latitude <P2and longitude O2; then the arcs HN = 90° - <PI,
PN = 90° - 4>2,and HP, which represents the distance between
~
218
PLANE
AND
SPHERICAL
TRIGONOMETRY
the points, form a spherical triangle called the terrestrial
The angle HNP
=
82
-
81, the difference
SPHERICAL TRIGONOMETRY
triangle.
of the longitudes,
219
its center. This sphere is called the celestial sphere, and the
center of the earth is taken as its center.
The location of a star, or any heavenly body, is the point where
a line drawn from the observer through the star pierces the celestial sphere. The location of one heavenly body with reference
to another is thus seen to depend upon the arcs of great circles
and spherical angles.
162. Fundamental
points, circles of reference, and systems
of coordinates.- The north pole P and the south pole Q of
the celestial sphere (Fig. 132) are the points where the earth's
axis produced pierces the surface of the sphere.
The horizon of any point on the earth is the great circle cut
from the celestial sphere by a horizontal plane through the
point. Thus HAH' is the horizon.
If at any point on the earth a perpendicular is erected to the
horizon at that point, the point where it pierces the celestial
sphere above the plane is called the zenith of the point, while
the piercing point below the plane is called the nadir of the point.
Thus, Z is the zenith and N the nadir of the point O.
The intersection of the plane of the earth's equator with the
celestial sphere is called the celestial equator.
Thus, EA W is
the celestial equator.
The great circle through the north pole and a star is called the
hour c11'cle of the star.
Ihus, PSi IS the hour circle of the star
at S.
The hour circle through the zenith is called the celestial
meridian, or simply the meridian.
Thus, PZEH is the celestial
meridian.
The hour angle of a star is the angle at the pole between the
meridian and the hour circle of the star. Thus, angle SPZ is the
hour angle of S. The hour angle is usually expressed as so many
hours, minutes, and seconds before or after noon. An hour
angle of 1 hr. is iT of 360°, or 15°.
The declination of a star is its angular distance north or south
of the equator.
The declination of the star S is IS.
The altitude of a star is its angular distance above the horizon
measured on the great circle through the zenith and the star.
The altitude of the star S is MS.
From the definition of latitude in Art. 159, it is easily seen that
the meridian arc EZ from the equator toward the zenith is the
latitude of the point on the earth's surface.'
is
known, as are also the arcs HN and PN. Two sides and the
included angle of the triangle HPN are then known and the triangle can be solved for the distance HP and for the bearing of one
point from the other.
The angle N HP is the bearing of P from H, and angle NP H the
bearing of H from P. The bearing will be represented by 'Y.
The bearing of a line is usually the smallest angle which the line
or path makes with the meridian through the point.
A complete solution of the terrestrial triangle by Napier's analogies and the sine law will furnish the bearings of any two points
and their distance apart.
EXERCISES
1. Find the shortest distance, in statute miles, between New York, 40°
45' N., 73° 58' W., and Chicago, 41 ° 50' N., 87° 35' W. Ans. 710 miles.
2. Find the shortest distance, in statute miles, between Chicago, 41 ° 50'
N., 87° 35' W., and San Francisco, 37° 40' N., 122° 28' W.
Ans. 1859 miles.
Find the shortest distance between the following places, and find the
bearing of each from the other:
3. New York, 40° 45' N., 73° 58' W., and Rio Janeiro, 22° 54' S., 43° 10' W.
Ans. Distance = 69° 47.8' = 4187.8 geographic miles.
Bearing of New York from Rio Janeiro, N. 24° 24.9' W.
Bearing of Rio Janelro from l'iew tork, S. i5U-10.5' 1:<;,
4. San Francisco, 37° 48' N., 122° 28' W., and Manila, 14° 36' N., 121 ° 5' E.
Ans. Distance = 100° 43.5' = 6043 geographic miles.
Bearing of San Francisco from Manila, N. 46° 3.4' E.
Bearing of Manila from San Francisco, N. 61 ° 51.7' W.
Find the shortest distance in statute miles between the following places
and check the work:
5. Chicago, 41° 50' N., 87° 35' W., and Manila, 14° 36' N., 121° 5' E.
6. Greenwich, 51 ° 29' N., and Valparaiso, 33° 2' S., 71° 42' W.
7. Paris, 48° 50' N., 2° 20' E., and Calcutta, 22° 35' N., 88° 27' E.
8. From a point at 40° N., 8° 15.6' W. a ship sails on the arc of a great
circle a distance of 3000 statute miles, starting in the direction S. 61 ° 15' W.
Find its latitude and longitude.
Ans. 12° 18.5' N., 46° 22.0' W.
9. What is the shortest distance on the surface of the earth from a point
A, 45° N., 74° W., and a point B which is 2000 miles directly west from A?
Ans. 1978 miles.
161. Applications to astronomy.-The
daily rotation of the
earth about its axis, from west to east, causes the stars to appear
to rotate daily from east to west. They move as if attached to
the surface of an immense sphere rotating about an axis through
..
l
220
PLANE AND SPHERICAL TRIGONOMETRY
SPHERICAL TRIGONOMETRY
The triangle ZPS is called the astronomical triangle.
ZP
is the colatitude of the observer, SZ is the coaltitude of the
star, SP is the codeclination of the star, and the angle SPZ is
the hour angle of the star. The answers to many practical problems of astronomy are obtained by solving the astronomical
triangle.
The determination of correct time, when the declination and altitude of the sun and the latitude of the observer are
To determine
221
LP, use [491],
{SiD.~8 - ~)
tan !p =
2
V
sm 8 sm
a = 64° 30'
log
b = 47° 40'
c = 74° 10'
= 186° 20'
28
sin (8 - c).
(8 - a)
sin 45° 30' = 9.85324
log sin 19° = 9.51264
colog sin 93° 10' = 0.00066
colog sin 28° 40' = 0.31902
8 = 93° 10'
8 - a = 28° 40'
8 - b = 45° 30'
8
C = 19°
,Zenith)
Z
log tan21P
log tan 1P
1P
P
-
.' . LP in hours
=
=
=
=
9.68556
9.84278
34° 50' 55/1
69° 41' 50/1
= 4 hr. 38 min. 47 sec.
Therefore, the time is 7 hr.
21 min. 13 sec. A.M.
Example 2.-Find
the time of
sunset for a place, latitude 32° 15'
N., when the declination is 17°
38' N.
Solution.-In
the triangle ZPS,
H
PS = 90° - 17° 38' = 72° 22'.
PZ = 90° - 32° 15' = 57° 45'.
ZS = 90°.
known, is obtained by solving for angle SPZ. The time of sunrise, neglecting refraction, may be found from the determination
of angle SPZ when the altitude is zero. We then have a quadrantal triangle, and its polar triangle is a
p
right spherical triangle.
Example I.-The latitude of Boston is
42° 20' N. A forenoon observation
showed the altitude of the sun to be 25°
30'. If the declination of the sun is 15°
z
50' N., find the time of observation.
Solution.-In
the triangle PZS,
s
FIG. 133.
PS = 90° - Dee!. = 74° 10'.
PZ = 90° - Lat.
ZS = 90° - Alt.
= 47° 40'.
= 64° 30'.
FIG. 134.
Triangle ZPS is then a quadrantal triangle which may be solved
by Napier's analogies.
A better method, however, is to solve
the polar triangle and from that obtain the required parts of the
given triangle.
In the polar triangle P'Z'S', LP' = 90°. LS' = 180° - PZ,
and LZ' = 180° - PS.
By Napier's rules or by (5) (Art. 136),
N
(Nadir)
FIG. 132.
cos S'Z' = cot S' cot Z'.
cos S'Z'
.
= cot (180°
-
57° 45') cot (180°
-
72° 22')
= tan 32° 15' tan 17° 38'.
log tan 32° 15' = 9.80000
log tan 17° 38' = 9.50223
log cos S'Z' = 9.30223
. . . S'Z' = 78° 25' 50"
.'. LP = 180° - 78° 25' 50" = 101° 34' 10".
And LP in hours = 6 hr. 46 min. 17 sec.
Therefore the sun sets at 6 hr. 46 min. 17 sec. P.M.
~
222
[21] tan 2(J
EXERCISES
1. A forenoon observation on the sun showed the altitude to be 41°. If
the latitude of the observer is 40° N. and the sun's declination is 20° N.,
find the time of observation.
Ans. 8 hr. 29 min. 20 sec.
2. Find the time of sunset at a place, latitude 45° 30' N., if the declination of the sun is 18° N.
Am. 7 hr. 17 min. 14 sec.
3. Find the altitude of the sun at 3 P.M. for a place, latitude 19° 25' N.
when the declination of the sun is 8° 23' N.
Ans. 45° 5'.
4. If the latitude of an observer is 8° 57' N., and the sun's declination is
23° 2' S., find the time of sunrise.
Ans. 6 hr. 15 min. 21.6 sec. A.M.
5. Find the time of sunrise at Chicago, latitude 41 ° 50', on June 21, when
the sun's declination is 23° 30'.
Ans. 4 hr. 28 min. 23 sec. A.M.
FORMULAS
+
[3] 1 + cot2 (J = csc2 (J.
and csc (J = ~.
[4] sin (J = ~
csc (J
sm (J
[5]
[6]
[7]
[8]
[9]
[10]
[11] G
=
-
sin (J).
[16] cos (a
[17] tan (a
[18] tan (a
(J.
= i:~1 + ;os
(3) = sin a cos {3 - COS a sin
-- (3) = COS a COS {3+ sin a sin
tan a + tan {3
+ fJ) = 1 - tan a tan {3.
a - tan fJ .
- fJ) = 1tan+ tan
a tan fJ
~
=
-!
cos (a
=~.
+ (3)+ !
cos (a
(Law of sines.)
- (3).
(Law of cosines.)
~
2 sm {3
[36] K = !ab sin 'Y.
a + b
tan !(a + (3)
1
[38] tan 2(a - (3)
[39] sin!a
2
1
[40] cos 2a
{3.
a-b
'\j
1
cot 2'Y.
m
= /(s -
b)(s
be
/s(s -- a) .
-- e).
= '\j
2
fJ.
2 sin2 0
=
be
/(s
b)(s - e).
[41] tan!a =
2
'\j s(s -- a)
1 = -, r where r =
[42] tan -a
[43] K
=
[44] s~n a
s-a
vs(s
-
s~n{3
sm a = sm
a)(s
u=
-- b)(s -
~
(S
-
a)(s
s
e).
s~n 'Y.
sm e
[45] cos a = cos b cos e + sin b sin e cos a.
[46] cos a
-
~
[35] K =
[
[19] sin 2(J = 2 sin (Jcos (J.
[20] cos 20 = cos2 0 - sin2 (J = 1
[23] cos ~O
i: ~1
+
(Area of segment.)
-
=
sm a
sm {3 sm 'Y
c2 - 2bc cos a.
[34] a2 = b2
b2 sin sin 'Y.
[14] cos (a + (3) = COSa COS{3 -- sin a sin fJ.
[15] sin (a
- 2COSo.
[22] sin ~(J
[33]
(J
lim
[12] sm (J < (J < tan (J. 8->0
1.
sin (J] =
[13] sin (a + (3) = sin a cos {3+ cos a sin {3.
.
2 tan (J
- tan2 i
1
[32] sin a sin {3 =
1
1
cos (J = sec (J and sec (J = -.
cos (J
1
1
tan (J = -'
cot (J and cot (J = tan
(J
sin (J
(}
tan
= -.
cas (J
cos tJ
cot (J = --;--'
sm (J
x = l cos (J. (Projection on x-axis.)
y = l sin (J. (Projection on y-axis.)
!r2«(J
=
223
sin (J .
[24] tan!(J = + 11 -- cos (J = 1 - cos (J =
2
- '\j 1 + cos (J
sin (J
1 + cos (J
[25] sin a + sin (3 = 2 sin Ha + (3)cos Ha - (3).
[26] sin a - sin (3 = 2 cos Ha + (3)sin Ha -- (3).
[27] cos a + cos (3 = 2 cos Ha + (3)cos Ha - (3).
[28] cos a - COS(3 = -2 sin Ha + (3)sin Ha - (3).
[29] sin a cos {3= ~ sin (a + (3) + ! sin (a - (3).
[30] cos a sin {3 = ! sin (a + (3) -- ! sin (a -- (3).
[31] cos a COS {3 = ! cos (a + (3)+ ! cos (a - (3).
[1] sin2 (J
cos2 (J = 1.
[2] 1 tan2 (J = sec2 (J.
+
TRIGONOMETRY
SPHERICAL
PLANE AND SPHERICAL TRIGONOMETRY
=
2 cos2 (J -: 1.
&
= -cos {3cos 'Y + sin {3sin cos a.
'Y
b)(s
- C).
~
224
AND SPHERICAL TRIGONOMETRY
PLANE
SPHERICAL
-. b).sin (8 - C)'where8=!(a+b+c).
sm
b' smc
8 sin (8 - a).
[48] cas !2a = '\j Isin sin
b sin c
. 1
[47] sm- a=
2
[49] tan
~
1
a
2
sin (8
= sm
.
-
r
(8-a
[50] cos ~a
2
=
=
[51] sin!a =
2
1
Icos (8
'\j
'\j
-
(8
~sin
a) sin
-: (3) C?S
~cos
8
~
-
b) sin (8
- c).
SIn 8
-
(8
sm (3sm l'
1- cos
USEFUL CONSTANTS
)' where
r
~8
1'),
where 8 = Ha + (3+ 1').
- ex).
sm (3sm l'
[52] tan 'j,a = R cos (8 - ex),where
.
R - I
- cos 8
'\jcos (8 - ex)cos (8 - (3)cos (8 - 1')
[53] tan Ha - b) = sin ~(ex- (3).
tan !c
sin Hex+ (3)
tan Hex - (3) sin !(a - b).
[54]
= sinHa+b)
coth
b)
tan Ha +
cos !~ex- (3).
[55J
= cos}(ex + 13)
tan k
tan Hex + (3) cos Ha - b).
[56]
= cosHa+b)
[57] cos!c sin Hex + (3) = cos h cos Ha - b).
[58] cos!c cos Hex + (3) = sin h cos Ha + b).
[59] sin!c sin Hex - (3) = cos h sin Ha - b).
[60] sin!c cos Ha - (3) = sin h sin !(a + b).
[61] ~
coth
=
7rr2E
180°'
[62] tan tE = vtan!8 tan!(8 - a) tan!(8 - b)tan!(8 - c)
[63] [r(cos 8
+ i sin
8)]n = rn[cos n8
(DeMoivre's
theorem,
ex3 a5
ex1
ex - - + - - - +
15
IT
ex2 ex4 a6
+ i sin
Art. 121.)
...
[64] sin ex
=
[65] cos ex
= 1- - + - - - + . . .
[66] tan ex = ex +
~
\2
14
ex3
"3
2ex5
~
+ 15 +
n8].
.
225
TRIGONOMETRY
1 cu. ft. of water weighs 62.5 lb. = 1000 oz. (Approx.)
1 gal. of water weighs 8t lb. (Approx.)
1 gal. = 231 cu. in. (by law of Congress).
1 bu. = 2150.42 cu. in. (by law of Congress).
(Approx.)
1 bu. = 1.2446 - cu. ft. = t cu. ft.
1 cu. ft. = 7t gal. (Approx.)
1 bbl. = 4.211- cu. ft.
1 m. = 39.37 in. (by law of Congress).
1 in.
=
25.4 mm.
1 ft. = 30.4801 cm.
1 kg. = 2.20462 lb.
1 g. = 15.432 gr.
lIb. (avoirdupois) = 453.5924277 g. = 0.45359+ kg.
lIb. (avoirdupois) = 7000 gr. (by law of Congress).
lIb. (apothecaries) = 5760 gr. (by law of Congress).
11. = 1.05668 qt. (liquid) = 0.90808 qt. (dry).
1 qt. (liquid) = 946.358 cc. = 0.946358 1., or cu. dm.
1 qt. (dry) = 1101.228 cc. = 1.101228 1., or cu. dm.
7r
= 3.14159265358979 = 3.1416 = 1H = 3t.
1 radian = 57° 17' 44.8" = 57.2957795°+.
1° = 0.01745329+ radian.
e = 2.718281828+, the base of the Napierian
(All approx.)
logarithms.
INDEX
Coordinates,
polar, 15
rectangular,
14
Corresponding
angles, 71
Cosine Theorem, 132, 202-,-204
Cycle of curve, 81
A
Abscissa, 15
Accuracy, 90
tests of, 91
Amplitude of curve, 81
Angle, addition and subtraction,
construction,
26, 31
definition of, 2
depression, 56
elevation, 56
functions of, 18
general, 12
generation of, 2
measurement
of, 4
negative, 2
of reflection, 102
of refraction, 102
positive, 2
\, rp~ of sphprieRltriH:l1~le, 215
Astronomical triangle, 220
B
Bearings,
D
3
57, 218
C
Celestial equator, 219
Celestial sphere, 219
Centesimal system, 4
Circular measure, 4
Complex number, 165
amplitude of, 172
division of, 170, 176
evolution of, 179
graphical representation
of, 167
involution of, 177
modulus of, 172
multiplication
of, 169, 176
polar form of, 172
DeMoivre's
Theorem,
177
E
Equations, trigonometric,
43, 71, 158
Exponential values of sin 0 and cos 0,
184
F
Forces, composition of, 93, 189
resuttantffl,
9&
Formulas, addition and suulradioll,
108
derivation of, for differences, 109
for sums, 108
for expansion of sin nO and cos nO,
182
for exponential
values of sin 0,
cos 0, and tan 0, 184
for powers of trigonometric
functions, 185
for products to sums, 122
for sums to products, 119
list of, 222
Functions, changes in value of, 78
hyperbolic, definition, 187
inverse, 42, 87
multiple-valued,
87
of angles greater than 90°, 62
of double angles, 114
of half angles, 117
period of, 81
227
~
228
PLANE
AND
SPHERICAL
Functions, periodic, 80
principal value of, 88
relations between, 34
single-valued,
87
trigonometric,
18, 28
TRIGONOMETRY
INDEX
0
Oblique triangles,
Ordinate, 15
130
P
G
Gauss's equations, 208
Graph, 79
mechanical construction
Periodic curve, 81
Polar triangles, 194
Projection, orthogonal,
92
of, 82
Q
H
Horizon, dip of, 95
Hour circle, 219
Hyperbolic functions, definition,
relations between, 188
I
Identity, 40
Index of refraction, 103
Infinity, 23
Inverse functions, 42, 87
principal values of, 88
Quadrantal
Quadrants,
triangles,
3
201
R
187
Radian, definition of, 4
Radius of circumscribed
circle, 131
of inscribed circle, 146
vector, 15
Reflection of light, 102
Refraction of light, 102
Right triangles, 47
S
L
Rpl'tor,
M
Meridian, 217
celestial, 219
principal, 217
Mil,5
Mollweide's equations,
ArPA of, 10, 911
,
uilier's Formula,
Light, reflection of, 102
refraction of, 102
Limit, 104
Lines, directed, 13
segments of, 13
Logarithms,
54
133
value of line, 13
Series for powers of trigonometric
functions, 185
Sexagesimal system, 4
Simple harmonic motion, 86
Sine Theorem, 130, 202
Solution,
of oblique spherical triangles, 210
of oblique triangles, 132
of right spherical triangles, 198
of right triangles, 47, 49, 54
Species, 198, 209
Spherical triangle, 193
oblique, 202
right, 195
N
Nadir, 219
Napier's analogies, 206
rules, 197
T
Tangent Theorem, 140
Terrestrial triangle, 217
Trigonometnc
curves, 79
Trigonometric
functions, applied to
right triangles, 28
calculation by computation,
20
by measurement,
20
changes in value of, 78
computation
of, 20, 184
definition of, 18
exponents of, 25
graph of, 79
inverse, 42, 87
line representation
of, 76
logarithmic, 19
natural, 19
of angles greater than 90°, 62
of complementary
angles, 30
of double angles, 114
of half angles, 117
229
Trigonometric
functions, natural, of
sums and differences of angles,
108
principal values of, 88
relations between, 34, 36
signs of, 19
table of, 24
transformation
of, 38
Trigonometric
ratios, 17
Trigonometric
series, 182
v
Vectors, 93, 171
composition of, 93
z
Zenith, 219
FIVE- PLACE
LOGARITHMIC
AND
TRIGONOMETRIC
WITH
TABLES
EXPLANATORY
CHAPTER
ARRANGED
CLAUDE
BY
IRWIN
Late Professor of Mathematics
Technology;
Author
PALMER
and Dean of Students,
Armour Institute
of a Series of Mathematics
Texts
AND
CHARLES
Professor
Emeritus
WILBER
LEIGH
of Analytic 1lfechanics, Armour Institute
A nthor of Prrwl1:col ,'1,[echonic.~
of Technology,
---
l
of
THIRD
FIFTH
McGRAW-HILL
NEW
EDITION
IMPRESSION
BOOK
YORK
AND
1935
COMPANY,
LONDON
INC.
~
PREFACE
TO THE THIRD EDITION
This new edition has given the author the opportunity to
substitute a number of new exercises which, it is hoped, will
prove helpful in introducing the student to the use of the tables.
CHARLES
CHICAGO, ILL.,
June,
1934.
COPYRIGHT.
1914,
MOGRAW-HILL
PRINTED
IN
THE
1925,
1935,
BY THE
BOOK COMPANY,
UNITED
STATES
OF
INO.
AMERICA
All rights reserved. This book, or
parts thereof,
may not be reproduced
in any form without permission of
the publishers.
THE
MAPLE
PRESS
COMPANY,
YORK,
PA.
ill
WILBER
LEIGH.
~
CONTENTS
LOGARITHMS
AND EXPLANATIONS
OF TABLES
PAGE
ART.
1.
Use
of
Logarithms.
2.
Exponents
3.
Definitions..
4.
Notation
5.
Systems
. . . . . . . . . . .. . . . . . . . . . . .. . . . . .. . . . . . . . . . . .
. . . . .. . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . .. . . . . .
. . . . .. . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .
of
6. Properties
Logarithms.
Rules
9.
The
to the
for
Tables
11.
To
12.
Rules
the
,........
"
. .. ., . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10...
Characteristic.
..... . . .. .. .. . . .. .. .. .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .
.. .. .. .. .. .. .. .... .... .. .. .. .. .. .. .. .... .. .. .. .. .
Find
the
for
Mantissa
Finding
of the
the
Find
15. Rules
Logarithm
Mantissa.
13. Finding the Logarithm
14. To
... . . . . . . . . . . . . . .
Base
Determining
Mantissa.
10.
..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
of Logarithms.
7. Logarithms
8.
..... .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
the Number
of a Number.
. .. .... .. .
of a Number
.... ......,..... .....
Corresponding
for Finding
the
Number
... .. . .. . . .
. . . . .. . . .. . . .. .. .
to a Logarithm
Corresponding
. . . . .. . . .
to a Given
Loga-
. 12
rithm
16.
To
Multiply
by
Means
of
Logarithms
... . . . . . . . . . . . . . . . . . . .
17. To Divide by Means of Logarithms
.. ...,.
.....
18. Cologarithms...............................................
19.
To
Find
the
Power
20.
To
Find
the
Root
of a Number
by
of a Number
by
Means
of Logarithms.
Means
..
of Logarithms
., . . .
.... .
21. Proportional Parts... ..
....................................
22.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
:Suggestiuns
23.
Changing
24.
Use
Systems
of
Table
II..
of
Logarithms..
..
., . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . .
25. Table III. Explanatory.....................................
26.
To
Find
27.
To
Find
Logarithmic
the
Angles
Function
Acute
Function
28.
Angle
of an
Acute
Corresponding
Angle..
to
a
Given
. . . . . . . . .. .. .
0
29. Functions
by
30.
of
Functions
and
90°.
Means
24
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
of Sand
Angles
Greater
T..
90?
Than
To
Find
the
Natural
33.
To
Find
the
Angle
34. Table V.
26
. . . . . . . . . . . . . . . . . . . . . . ..
27
29
Function
of an
Corresponding
to
. . . . . . . . . . . . . . . . . . ..
a Given
Natural
Function.
..
I.
Logarithms
Table
II.
Table
III.
Lcgarithms
Table
IV.
Natural
Table
V.
Radian
........ . . . . . . . . . . . . . . . . ..
of Numbers
Conversion
of Logarithms...
of Trigonometric
Trigonometric
Measure.
29
30
31
of Interpolation
Table VI.
Angle.
Explanatory
Table
26
, . . . . . . . . . . . . . . .. .. .. ...
31. Table IV. Explanatory
32.
13
14
14
15
15
16
l(j
20
21
23
23
Logarithmic
. . . . .. . . . . .. . . . . . . . . . . . . . . ..
near
35. Errors
1
1
2
2
3
3
4
6
7
7
8
9
10
10
35
. . . . . . . . . . . . . . . . . . . . . ..
56
Functions..
Functions...
. . . . . . . . . ..
., . . . . ., . . . . . . ..
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Constants and Their Logarithms. . . . . . . . . . . . . . . . . . ..
v
32
., . . . . . . . . . . . . . . . . . . . . . ..
57
111
135
136
l
LOGARITHMS
AND EXPLANATION
OF TABLES
1. Use of logarithms.-By
the use of logarithms, the processes
of inultiplication, division, raising to a power, and extracting a
root of arithmetical numbers are usually much simplified. The
process of multiplication is replaced by one of addition, that of
division by one of subtraction, that of raising to a power by a
simple multiplication, and that of extracting a root by a division.
Many calculations that are difficult or impossible by other
mathematical methods are readily carried out by means of logarithms.
It was said by the great French astronomer, Laplace,
that the method of logarithms, by reducing to a few days the
labors of many months, doubled, as it were, the life of an astronomer, besides freeing him from the errors and disgust inseparable
from long calculations.
Of course, these same advantages are
shared by others who find it necessary to perform numerical
calculations. *
2. Exponents.-The
student is already familiar with the following definitions and theorems from algebra, concerning exponents. For convenience they are restated here.
Definitions.
(1)
(2)
(3)
(4)
. an
= a
. a . a . . . to n factors. n an integer.
a-n = -,1
an
aO = 1.
n
am = Van.
Theorems.
(1)
an . am = an+m.
(2)
an --;-am = an-m.
(3) (a. b . e . . . r)n = an. bn . en . . . rn.
(4)
(5)
Gr =
(an)m
~n'
= anm.
* "The miraculous powers of modern calculations are due to three inventions: the Hindu Notation, Decimal Fractions, and Logarithms."
(CAJORI,
"A History of Elementary
Mathematics.")
1
l
PLANE TRIGONOMETRY
2
3. Definitions.-If
that
LOGARITHMS
three numbers N, b, and x have such values
N = bx,
then x is called the logarithm * of N to the base b. In words, this
gives the following.
DEFINITION.-The logarithm of a number to a given base is the
exponent by which the base must be affected to produce that number.
If, in the equation N = bX, all possible positive values are given
to N, while b is some positive number other than 1, the corresponding values of x form a system of logarithms.
4. Notation.-If
4 is taken as a base, then, in the language,
or notation, of exponents,
43
= 64.
In the language, or notation, of logarithms, the same idea is
expressed by saying the logarithm of 64 to the base 4 is 3. This
is abbreviated and written
10g4 64 = 3.
EXPONENT
LOGARITHMIC NOTATION
NOTATION
4. =
34 =
53 =
4°.. =
161 =
103 =
10g4 1024
1024.
81.
125.
2.
64.
1000.
= 5.
10g381 =
log. 125 =
log4 2 =
10g1664 =
log,u 1000 =
4.
3.
0.5.
t.
3.
EXERCISES
Answer as many as possible orally.
Express in logarithmic notation:
4. 64°.6 = 8.
5. 104 = 10,000.
6. 1251 = 5.
1. 3' = 27.
2. 5' = 125.
3. 7' = 343.
Express
in exponent
10. log.
256
8. 102.3979 = 250.
9. 101.64.6 = 35.2.
notation:
12. log,. 2 = 0.25.
= 8.
11. log. 216 = 3.
Find the logarithms
16. log. 36.
17. log, 243.
18. log. 3125.
7. 32°.4= 4.
13. log,o 643 = 2.8082.
of the
19.
20.
21.
* The word "logarithm"
meaning number.
following:
log9 729.
logs 512.
loglO 100,000.
14. log,o 429 = 2.6325.
15. log,o 99.9
=
1.9996.
22. logs 2.
23. log,. 128.
24. log,o 0.001.
is derived from logos meaning
ratio and arithrrws
AND EXPLANATION
Find the value of x in the following:
29. log,o x = -3.
25. log, x = 4.
30. log, x = -2.
26. log,o x = 4.
27. log,. x = t.
31. log,. x = 0.75.
32. logs x = t.
28. log,o x = O.
OF TABLES
3
33. log.. x = R.
34. log,. x = ~.
35. log,.. x = i.
36. log49x = t.
Find the value of x in the following:
45. log, 27 = 0.75.
37. log, 1000 = 3.
41. log, 8 = 0.5.
46. log, 2 = 0.125
42. log, 4 = 0.25.
38. log,
81 = 4.
43. log, 16 = t.
39. log, 256 = 4.
47. log, 36 = i.
48. log, 100 = j.
40. log, 1024 = 10. 44. log, 18 = 0.5.
49. What are the logarithms of 2, 4, 8, 16, 32, 64, 128, 256, to the base 21
50. What are the logarithms of 2, 1, !, I, 1, h, -h, -h, ns, to the base 21
51. What are the logarithms of 3, 9, 27, 81, 243, 729, to the base 3? To
the base!?
52. What are the logarithms of 3, 1, !, !, -l." .IT, 1!!.' to the base 3? To
the base!?
5.. Systems of logarithms. *-While, theoretically, any positive
number other than 1 may be used as a base for a system of logarithms, in practice only two bases are used.
(1) The common system, or Briggs' system, of which the base
is 10.
(2) The natural system, also called the hyperbolic, or Napierian
system, of which the base is a number that to seven decimal
plaf~PS is 2.7182818. This base is usually represented by the
letter e.
The common system is the one commonly used in computing,
and the natural system in more advanced and theoretical work.
6. Properties of logarithms.-The
use of logarithms depends
upon the following properties, which are true for any base greater
than unity:
(1) The logarithm of 1 is zero.
Since
bO
=
1, for any base, 10gb 1
= O.
(2) The logarithm of the base of any system is unity.
Since bl = b, for any base, 10gbb = 1.
Logarithms
were invented by John Napier, Baron of Merchiston, of
*
Scotland, who lived from 1550 to 1617. They were described by him jn 1614.
A contemporary
of Napier,
Henry Briggs (1556-1631),
professor
of
Gresham College, London, modified the new invention by using the base 10,
and so made it more convenient
for practical
purposes.
(See CAJORI,
"A History of Elementary
Mathematics,"
p. 160 et seq. For a very complete account of logarithms, see CAJORI, FLORIAN, "History of the Logarithmic and Exponential
Concepts.")
~
4
PLANE
LOGARITHMS
TRIGONOMETRY
10-2
10-3
10-4
10-5
If N = b'" and M = bY, then, by the definition of a logarithm,
10gb N = x and 10gb M = y. We have also by the definitions
and theorems of exponents and logarithms:
(a) N X M = b'" X bY
= b"'+Y.
, . . 10gb (N X M)
= x + y = 10gbN + 10gbM.
(b) N + M = b'"+ bY
= b"'-Y,
, . , 10gb (N
+ M) = x - y = 10gbN - 10gbM.
(c) Nn = (b",)n = bn",. . . . 10gb (Nn)
= nx = n 10gbN.
(d) VN
1
0.01
0.001
0.0001
0.00001
OF TABLES
5
. . . log 0.01 = -2 = 8 - 10.
. . . log 0.001 = -3 = 7 - 10.
. . . log 0.0001 = -4 = 6 - 10.
. . . log 0.00001 = -5 = 5 - 10.
It is evident that these are the only numbers between 0.00001
and 100,000 which have integers for logarithms.
Every other'
number in this range has, then, for a logarithm an integer plus
or minus a fraction.
This fraction is put in the form of a decimal.
For instance, the logarithm of any number between 1000 and
10,000 is between 3 and 4, or it is 3 + a decimal.
For a number
between 100 and 1000 the logarithm is 2 + a decimal.
Between
0.01 and 0.1, the logarithm may be -2 + a decimal or -1 - a
decimal; but, in order that the fractional part of the logarithm
may always be positive, it is agreed to take the logarithm so that
the integral part only is negative.
Usually, then, the logarithm of a number consists of two parts,
an integer and a fraction, the fraction being the approximate
value of an irrational number.
The integral part is called the characteristic.
The fractional part is called the mantissa.
The logarithm is the characteristic plus the mantissa.
The mantissas of the positive numbers arranged in order are
called a table of logarithms.
The logarithm of 3467 consists of the characteristic 3 plus some
mantissa, because 3467 lies between 1000 and 10,000. The logarithm of 59,436 is 4 + a decimal, because 59,436 lies between
10,000 and 100,000. The log 0.0236 = -2 + a decimal, because
0.0236 lies between 0.01 and 0.1.
It is readily seen that multiplying a number by IOn increases
the characteristic by n, where n is an integer; and dividing a
number by IOn decreases the characteristic by n, for
log (N X IOn) = log N + log IOn = log N + n log 10
= log N + n, and
log (N + IOn) = log N - log IOn = log N - n log 10
= log N - n.
1
_n/1
1
.
= (b"');; b;;"'. .'. 10gbV
N = -x = - 10gbN.
=
n
n
The following theorems are, therefore, established:
(3) The logarithm of a product equals the sum of the logarithms
of the factors. By (a).
(4) The logarithm of a quotient equals the logarithm of the dividend
minus the logarithm of the divisor. By (b).
0
(6) The logarithm of a root of a number equals the logarithm of
the number divided by the index of the root. By (d).
The truth of the statements in Art. 1 follows from these theorems. That is, the process of multiplication is replaced by an
addition; division by a subtraction; raising to a power, by a
multiplication; and extracting a root, by a division.
7. Logarithms to the base lO.-In
what follows, if no base
is stated, it is understood that the base 10 is used.
When the base is 10 we evidently have the following:
105 = 100,000
104 = 10,000
103 = 1000
102 = 100
101 = 10
10° = 1
10-1 = 0.1
=
=
=
=
AND EXPLANATION
. . . log 100,000 = 5.
. . . 10,000 = 4.
. . . log 1000 = 3.
. . . log 100 = 2.
. . . log 10 = 1.
. . . log 1 = O.
. . . iog O. 1 = -1 = 9 - 10.
This establishes th'3 following:
THEoREM.-The
position of the decimal point in the number
affects the characteristic of the logarithm only, the mantissa remaining unchanged for the same sequence of figures.
A
l
6
LOGARITHMS
PLANE TRIGONOMETRY
If log 31
The advantages in using the base 10 are that the characteristic
can be determined by inspection, and that the mantissa remains
unchanged for the same sequence of figures.
Thus,
log 934,700 =
log 9347 =
log 9.347 =
log 0.009347 =
5.97067.
3.97067.
0.97067.
In computations involving negative characteristics, to avoid
the use of the negative, 10 is usually added to the characteristic
and subtracted at the right of the mantissa.
In writing logarithms in this form, the characteristic,when 10 is added, is 9 minus
the number of zeros immediately at the right of the decimal point.
in the above,
log 0.009347
= 7.97067
-
10.
The characteristics of the following are as given: of 3426 is 3 by
rule (1); of 3.2364 is 0 by rule (1); of 0.00639 is -3, or 7 - 10,
by rule (2); of 2.04 is 0 by rule (1); of 0.000067 is -5, or 5 - 10,
by rule (2).
4. 8.004.
3. 1.111.
8. 0.00046.
9. 0.000395.
6. 80004.
10. 0.04762.
13. 3.004.
14. 2525.1.
16. 1000.25.
7
give:
24. log 0.031.
26. log 0.0031.
26. log 0.000031.
27. log 0.3100.
28. log (31 X 107).
29. log (31 X 10-5).
* Professor Briggs' tables were computed
to 14 places, but were not
finished by him.
They were completed by Adrian Vlacq (1628), who shortened them to 10 places, and finished a table including the numbers from 1 to
100,000.
Briggs' and Vlacq's tables are essentially the same as those now in use.
They have been checked and recomputed in part many times.
At present,
errors found in tables are typographical.
The most complete check was
undertaken
by the French authorities
in 1784. It required the labors of
nearly a hundred mathematicians
and computers for over two years.
They
computed to 14 places the logarithms
of all integers from 1 to 200,000,
besides natural and logarithmic trigonometric
functions.
These tables were
never printed.
Two manuscript copies are preserved.
EXERCISES
In each of the following state the characteristic
1. 923.
6. 42,376.
11. 32.54.
2. 425.03.
7. 3.2067.
12. 0.0123.
1.49136,
OF TABLES
9. The mantissa.-The
determination of the decimal part of
a logarithm, the mantissa, is more difficult than the determination of the characteristic.
Because of this difficulty the mantissas have been carefully determined and arranged in tables of
logarithms. * They are given to three, four, five, or more places
of decimals.
The degree of accuracy in computations made by logarithms
depends upon the number of places in the table used; the more
places in the table the greater the degree of accuracy.
The tables
generally used are those having from four to seven places.
10. Tables.-Upon
examining a five-place table of logarithms
(see Table I), it is noticed that the first column has the letter N
at the top and the bottom.
This is an abbreviation for number.
The other columns have at top and bottom the numbers 0, 1, 2, 3,
. . . 9. Table I contains the integers from 1000 to 11,009.
Pages 36 to 53 have the numbers from 1000 to 10,009. Here the
first three figures are printed in the column marked N and the
fourth figure at the top and bottom of another column. Thus, to
locate 4756, 475 is found in the N-column on page 43 and 6 in
thp column headpd 6. Page~ ,r)4and 5,15contai
10,000 to 11,009, where the first four figures are printed in the
N-column.
The columns of numbers after the first consist of the mantissas
of the numbers located in the N-column and at the top, or bottom,
of another column. These mantissas are printed correct to five
decimals, except on pages 54 and 55, where they are given to
seven places. To save space, the first two figures of the mantissas
3.97067 = 7.97067 - 10.
8. Rules for determining the characteristic.-From
the foregoing considerations the following rules for determining the
characteristic are evident:
(1) When the number is greater than 1, the characteristic is positive, and is one less than the number of digits to the left of the decimal
point.
(2) When the number is less than 1 and expressed decimally, the
characteristic is negative, and is one more than the number of zeros
immediately at the right of the decimal point.
When the characteristic is negative the minus sign is placed
above the characteristic to show that it alone is negative.
Thus,
in log 0.009347 = 3.97067, the 3.97067 means -3 + 0.97067.
It should not be written -3.97067, for then the minus sign would
indicate that both characteristic and mantissa were negative,
while we have agreed that the mantissa shall always be considered
Thus,
=
21. log 310.
22. log 31,000.
23. log 3.1.
AND EXPLANATION
to the base 10:
16. 89,236.
17. 0.2146.
18. 333.33.
19. 105 X 2.
20. 10-3 X 6.
.
-------
l
8
PLANE
TRIGONOMETRY
are printed in the O-column only. Any such two figures go with
the other figures to the right and below until another two figures
is found in the O-column. Except that when an asterisk (*) is
found before the three figures given in the other columns, the
first two figures of the mantissa are taken from the next line
below.
When a mantissa ends in a figure 5 it is printed 5 when it is
really less than printed; otherwise, a mantissa when ending in 5
is larger than printed.
Thus, if the mantissa is 0.0273496, in contracting it to five
places, it is printed 0.02735. This is to guide one wishing to write
the mantissas correct to four places.
For the meaning of the Prop. Parts, see Art. 21. For the
meaning of the numbers at the foot of the pages and connected
with Sand T, see Art. 29.
Notice that, when advancing in the table, the mantissas
increase.
The difference between two consecutive mantissas is called the
tabular difference.
11. To find the mantissa of the logarithm of a number.-Use
Table I, pages 36 to 53.
(1) When the number consists of four sigmjicant figures.
Example.-Find
thelllaI1ti~sa of log 4673.
j
-"
- -
-
-
-
. --
-
OF TABLES
9
Thus, Mant. of log 4.78 = Mant. of log 4780 = 0.67943.
Mant. of log 39 = Mant. of log 3900 = 0.59106.
Mant. of log 4 = Mant. of log 4000 = 0.60206.
(3) When the number consists of five or more significant figures.
Example I.-Find
the mantissa of log 39,467.
Since 39,467 lies between 39,460 and 39,470, its mantissa must
lie between the mantissas of these numbers.
Mant. of log 39,460 = 0.59616.
Mant. of log 39,470 = 0.59627.
The difference between these mantissas is 0.00011, which is the
tabular difference.
Since an increase of 10 in the number increases the mantissa 0.00011, an increase of 7 in the number will
increase the mantissa 0.7 as much, or the increase is
0.00011 X 0.7 = 0.000077 or 0.00008.
+ 0.00008 = 0.59624.
. . . Mant. of log 39,467 = 0.59616
The process of finding the
polation. As carried out, it
logarithm is proportional to
assumption is not strictly true
mantissa as above is called interis assumed that the increase of the
the increase of the number. This
as will be seen in Art. 35.
Example 2.-Find the mantissa of log 792,836.
Mant. of log 792,900_~Q._8~~22
')
and the 3 at the top of the page. The mantissa of log 4673 is
found to the right of 467 and in the column headed 3.
. . . mantissa of log 4673 = 0.66960.
Tabular difference = 0.00006
Since an increase of 100 in the number increases the mantissa
0.00006, an increase of 36 in the number increases the mantissa
0.00006 X 0.36 = 0.00002,
In like manner find the following:
Mant. oflog 4799
Mant. of log 23.78
Mant. of log 2.955
Mant. of log 0.0003964
Mant. of log 3560
Mant. of log 4930
Mant. of log 556,700
Mant. of log 0.001001
LOGAR/TH MS AND EXPLANATION
=
=
=
=
=
=
=
=
0.68115.
0.37621.
0.47056.
0.59813.
0.55145.
0.69285.
0.74562.
0.00043.
(2) When the number consists of one, two, or three significant
figures. The number is found in the N -column and the mantissa
to the right in the O-column.
correct to the nearest fifth decimal place.
.' . Mant. of log 792,836 = 0.89916 + 0.00002 = 0.89918.
These processes should seem reasonable; but, since they are
to be performed so frequently, it is best to work by rule.
12. Rules for finding the mantissa.-(I)
For a number consisting of four figures, find the first three figures of the number in the
N-column and the fourth figure at the head of a column; then read
the mantissa in the column under the last figure and at the right of
the first three figures.
(2) For a number consisting of one, two, or three figures, find the
number in the N-column and the mantissa to the right in the column
headed O.
~
10
LOGARITHMS
PLANE TRIGONOMETRY
Example 2.-Find the logarithm of 3.4676.
Rule (1) for characteristic gives o.
Rule (3) for mantissa gives 0.54003.
Example 3.-Find
Rule
= 0.54003.
3.4676
the logarithm
(2) for characteristic
gives
of 0.00039724.
A change in the characteristic
mal pomt.
Thus,
EXERCISES
10. log 0.492357 = 1.69228.
11.
12.
13.
14.
16.
16.
17.
18.
log
log
log
log
log
log
log
log
11
. . . 2.58939 = log 388.5.
4.
Rule (3) for mantissa gives 0.59905.
. . . lo!! 0.00039724 = 4.5990fi = 6.fi9905 - 10.
Verify the following by the tables:
1. log 9376 = 3.97202.
2. log 4.236 = 0.62696.
3. log 220 = 2.34242.
4. log 1.11 = 0.04532.
6. log 20 = 1.30103.
6. log 0.02 = 2.30103.
7. log 0.00263 = 7.41996 - 10.
8. log 26.436 = 1.42220.
9. log 3.1416 = 0.49715.
OF TABLES
In nearly every problem involving logarithms, it is not only
necessary to find the logarithms of numbers; but the inverse
process, that of finding the number corresponding to a logarithm,
has to be performed.
Since the position of the decimal point in no way affects the
mantissa, we should expect to determine the sequence of figures
in the number from the mantissa.
And since a change in the
position of the decimal point increases or decreases the characteristic, the decimal point may be located in the number when the
characteristic of the logarithm of the number is known.
(1) When the mantissa of the given logarithm is given exactly in
the table.
Example.-Find
the number having 2.58939 for its logarithm.
Find in the table the mantissa 0.58939. To the left of this mantissa, in the N-column, find the first three figures, 388, of the
number, and at the head of the page find the fourth figure, 5, of the
number.
The number then consists of the sequence of figures
3885, but we do not know where the decimal point is until we
consider the characteristic, which is 2. Hence there must be
three figures at the left of the decimal point.
(3) For a number consisting of more than four figures, find
the mantissa for the .first four figures by rule (1) and add to this the
product of the tabular difference by the remaining figures of the
number considered as a decimal number.
13. Finding the logarithm of a number.-In
finding the
logarithm of a number, it is best to determine the characteristic
first and then look up the mantissa.
Perform all the interpolations without the aid of a pencil if possible. The use of the proportional parts is explained in Art. 21; but the student is
advised to become familiar with interpolating without their help.
Example I.-Find
the logarithm of 92.36.
The characteristic is 1, by rule (1) for characteristics.
The
mantissa is 0.96548, by rule (1) for mantissas.
. . . log 92.36 = 1.96548.
. . . log
AND EXPLANATION
276.392 = 2.44153.
0.027646 = 8.44164 - 10.
0.0049643 = 3.69586.
0.029896 = 2.47561.
2.71828 = 0.43429.
99.999 = 2.00000.
1111.11 = 3.04575.
100.03 = 2.00013.
changes the location of the deci-
4.58939 = log 38,850,
2.58939 = log 0.03885,
7.58939 - 10 = log 0.003885.
(2) When the mantissa of the given logarithm is not given exactly
in the table. In this case two other consecutive mantissas can
always be found between which the mantissa of the given logarithm lies. The number of four figures corresponding to the
smaller of these mantissas gives the first four figures of the number
sought. The fifth, and often the sixth, figure can then be found
by interpolating,
assuming that, for comparatively
small differences in the numbers, the differences in the numbers are proportional to the differences in the logarithms of the numbers.
For using the proportional
parts in interpolating,
see
Art. 21.
Example.-Find
the number whose logarithm is 1.49863.
14. To find the number corresponding to a logarithm.-If
log
31.416 = 1.49715, then 31.416 is the number corresponding to
the logarithm 1.49715. It is sometimes called the antilogarithm
and is written 31.416 = log-l 1.49715.
a
~
12
PLANE TRIGONOMETRY
In the table find the mantissas 0.49859
which the given mantissa lies. Thinking
of figures in the numbers,
0.49872 = Mant. of log
0.49859 = Mant. of log
and 0.49872, between
only of the sequence
3153
3152
0.00013
1
Hence a difference of 0.00013 in the logarithm makes a difference
of 1 in the number. Now the given mantissa is 0.00004 larger
than the smaller one. Then the number having 0.49863 as the
mantissa of its logarithm is
0.00004
4
X 1 =
13 = 0.3
0.00013
larger than 3152. Hence, the sequence of figures for the number
having 1.49863 as a logarithm is 31,523. Since the characteristic
is 1,
1.49863 = log 31.523.
The interpolation should be carried out mentally, leaving out
zeros and taking i\- X 1 = 0.3.
This could also be stated as a proportion,
13: 4 = 1: x, .'. x = 0.3.
15. Rules for finding the number corresponding to a given
logarithm.-(I)
When the mantissa of the g1:venlogarithm is found
tft-v
t,.\.t.u.t.\."
t-,!"\.,
Jt.,
-.'t-
t-""
\.t.'
t...
J":J
'.".'
'~J
""-,,
AND EXPLANATION
LOGARITHMS
,."...",.v,-"'
'--'
J v
TABLES
13
EXERCISES
Find the values of x or verify the following:
11. 8.12112 - 10 = log 0.013217.
1. 3.70944 = log 5122.
12. 6.28697
2. 2.58377 = log 0.03835.
= log x.
13. 6.89909 = log x.
3. 1.74819 = log x.
4. 7.94236 - 10 = log 0.0087U7. 14. 11.46729 = log x.
16. 9.92867 - 20 = log x.
6. 0.47712 = log x.
16. 3.88888 = log x.
6. 3.47954 = log 3016.7.
17. 3.33333 = log x.
7. 2.57351 = log 374.55.
18. 4.0002565 = log 10005.9.
8. 0.92876 = log x.
19. 2.0331894 = log 107.9417.
9. 9.23465 - 10 = log x.
20. 3.0275278 = log 1065.437.
10. 4.92317 = log 0.00083786.
For exercises 18 to 20 use pages 54 and 55 of Table I.
16. To multiply by means of logarithms.-Property
(4) of
Art. 6 gives the following:
RULE.-To find the product of two or more factors, find the sum of
the logarithms of the factors,. the product is the number corresponding
to this sum.
Example I.-Find
the product of 34.796 X 0.0294 X 3.1416.
log 34.796 = 1.54153
log 0.0294 = 8.46835
log 3.1416 = 0.49715
Process.
-
10
log of proouct = O}i0703
. . . pl'uJud = 3.2139
t-,,,,
the left of the mantissa in the N -column, and the fourth figure is at
the head of the column in which the mantissa is found.
(2) When the mantissa of the given logarithm is not found exactly
in the table, find the mantissa nearest the given mantissa but smaller.
The first four figures of the number are those corresponding to this
mantissa, and are found by rule (1). For another figure, divide the
difference between the mantissa found and the given mantissa by the
tabular difference.
In both (1) and (2), place the decimal point so that the rules for
determining the characteristic may be applied and give the characteristic of the logarithm.
Example.-Find
the number corresponding to 3.87626.
Mantissa nearest 0.87626 is 0.87622 = Mant. of log 7520.
Tabular difference = 6.
Difference between mantissas = 4.
4 + 6 = 0.7 to nearest tenth.
.'. 3.87626 = log 7520.7.
OF
Example 2.-Find
( -0.004682).
Process.
the
product
of 3.276 X (-4.6243)
X
log 3.276 = 0.51534
log 4.6243 = 0.66505n
log 0.004682 = 3.67043n
log of product = 2.85082
. . . product = 0.070928
Since the logarithms cannot take into account the negative
numbers, the easiest way to keep count of the negative factors is
to place a letter n after their logarithms.
In finding a sum or
difference of logarithms, write an n after the result only if an
odd number of the separate logarithms are affected by an n.
This method was introduced
by the great mathematician
Gauss (1777-1855).
---..-------
--------------------------------------------------------------------------------
~
14
TRIGONOMETRY
PLANE
LOGARITHMS
17. To divide by means of logarithms.-Property
(5) of Art. 6
gives the following:
RULE.-To find the quotient of two numbers, subtract the logarithm
of the divisor from the logarithm of the dividend; the quotient is the
number corresponding to this difference.
Example I.-Find
the quotient of 27.634 -+ 5.427.
Process.
log 27.634
log 5.427
log of quotient
=
=
=
Thus, colog 9.423 = log 1
1.44144
0.73456
-
15
log 9.423.
log 7.246 = 0.86010
log 0.8964
0.70688
4.27 X 0.3987 X 27.89
Here find logarithm of the numerator then that
nator.
Process.
log 7.246 = 0.86010
log 4.27 =
log 0.8964 = 9.95250- 10 log 0.3987
=
log 5.463 = 0.73743
log 27.89 =
log of Num. =' 1.55003
log of Den. =
log of Den. = 1.67653
log of quotient
= 9.87350 - 10
.'. quotient = 0.74732
the arithmetical
OF TABLES
EXPLANATION
log 1 = 10.00000 - 10
log 9.423 = 0.97419
. . . colog9.423 = 9.02581 - 10
The solution of Example 2 (Art. 17) takes the following form;
= 9.95250
log 5.463 = 0.73743
. . . quotient = 5.0919
Example 2.-Evaluate 7.246 X 0.8964 X 5.463 .
18. Cologarithms.--=-The
AND
colog 4.27 = 9.36957
= 0.39935
-
10
-
10
colog 0.3987
colog 27.89 = 8.55455 - 10
log of quotient = 9.87350 - 10
.'. quotient = 0.74732
19. To find the power of a number by means of logarithms.Property (6) of Art. 6 gives the following:
of the denomi-
0.63043
9.60065 - 10
1.44545
1.67653
RULE.-To
find the power of a number, multiply the logarithm
of the number by the exponent of the power; the number corresponding
to this logarithm is the required power.
Example I.-Find
Process.
Iogarif.hm
the value of (2.378)6.
log 2.378 = 0.37621
6 X log 2.378 = 2.25726 = log (2.37S)6
complement.
Since the reciprocal of N is
Alsolog .z; = log M
~,colog N
+ log ~
= log M
= log
Example 2.-Find
~ = log 1 -
+ colog
logN.
Process.
log 237.45 = 2.37557
t
N, that is:
the value of (237.45)t
X log 237.45
= 1.69684
= log (237.45)t
.'. (237.45)t = 49.756
20. To find the root of a number by means of logarithms.Property (7) of Art. 6 gives the following:
RULE.-To find the root of a number, divide the logarithm of the
number by the index of the root; the number corresponding to this
logarithm is the root required.
The logarithm of the quotient of two numbers is equal to the logarithm of the dividend plus the cologarithm of the divisor.
That is, subtracting the logarithm of a number is the same as
adding the cologarithm of the number.
It is evident that, by
using cologarithms, the work can often be made more compact
than otherwise.
It should be noted that it is never necessary to
use cologarithms.
To find the cologarithm of a number, subtract the logarithm of
the number from 10 - 10. Do the work mentally, beginning at
the left and subtracting each figure from 9, except the last
significant figure at the right, which is to be taken from 10.
Example I.-Find
~27.658.
log 27.658 = 1.44182
Process.
t
A
X log 27.658 = O.28S36 = log ~27.658
. '. ~27.658 = 1.9425
~
16
PLANE TRIGONOMETRY
Example 2.-Find
{l0.008673.
Process.
log 0.008673 = 7.93817 log {l0.008673 = H7.93817
= H57.93817
= 9.65636 . '. {Io.OO86r3 = 0.45327
LOGARITHMS AND EXPLANATION
10
- 10)
- 60)
10
to oivide Rome number
by 35.
17
In writing numbers correct to a certain number of figures, take
in the last place the figure that is nearest the true result when this
is possible. If the next figure after the last one to be taken is
5 followed only by zeros, most computers take the nearest even
figure for the last one. Thus, if the number is 0.02467500
it would be taken 0.02468 to five places; and if 0.02468500
it would also be taken 0.02468.
In working with tables, use the pencil as little as possible. Work
for accuracy first and then for speed.
Write out a scheme for all logarithmic work before referring
to the table. Be sure that your work is arranged so that it could
be followed at any time by yourself or another person.
Example.-Write
out a scheme for finding the value of
Remark.- When a logarithm with a negative characteristic
is to be divided by a number not exactly contained in the characteristic, it is best first to add and subtract such a number of times
10 that, after dividing, there will be a minus 10 at the right. In
the above, before dividing (7.93817 - 10) by 6,50 was added and
subtracted.
If the divisor had been 3, however, the division
could have been performed by writing the logarithm in the form
3.93817 and dividing at once by 3.
21. Proportional
parts.-In
Table I, the marginal tables,
marked Prop. Parts, contain the products of the tabular differences by 1, 2,3, . . . 9 tenths.
These products are arranged for
convenience in interpolating.
The work should be done mentally.
Thus, in interpolating, if the tabular difference is
1 315
35, then the marginal table is as given. In finding the
~ Ih:g
logarithm, it is required to multiply, say, 35, by some
~ ~~:g
number; and, in finding a number corresponding to a
~
~l:g
1o!!:flrithm, it iR requirpo
OF TABLES
x =
9.46 X (41.6)2 X V9.462 .
276.2 X 3.4675
Scheme.
log 9.46
=
2 log 41.6 =
!
log 9.462 =
colog 276.2 =
colog 3.4675 =
log x =
x=
g ~:~
(1) MultIply 3b by
EXERCISES
Solve by logarithms:
1. 32.758 X 8.3759.
2. 9.0083 X 0.072893.
3. (-0.001009)
X 52456.7.
4. 64.785 X 5.6346 X 0.01025.
5. 2.71828 X 1000 X 0.31461.
6. 59.7642 -;- 5.73894.
7. 0.083467 -;- 0.0046834.
8. 11.01101 -;- 96.15.
9. 579.996 -;- (-37.16).
10. 9.94923 + 429.693.
11. (1.74)17.
12. (4.43769)3 X 0.9746.
13. (1.5651)t.
14. {II7.
15. V77 + ~.
16. ~0.0000067.
17. (1.42) 12.
35 X 0.6 = 21.0
35 X 0.08 = 2.8
. . . 35 X 0.68 = 23.8
(2) Divide 29 by 35.
Process. Dividend..
29
giving 0.8
Next less
28.0
Remainder...
10
Next less
7.0 giving 0.02
Remainder...
Next less. . . .
28 giving 0.008
Etc.
0.828 . . . = quotient.
Process.
w
In interpolating, the division is usually only to determine the
nearest first figure, and therefore can easily be done mentally.
22. Suggestions.-In
interpolating,
do not carry logarithms
beyond the number of decimal places given in the table.
18 .
--
]I(
} 10.002396
''J 8.926 .
Ans.
274.37.
0.65664.
Ans.
-52.929.
Ans.
3.7416.
Ans. 855.2.
Ans.
10.414.
Ans.
17.822.
Ans.
0.11452.
Ans.
-15.608.
Ans.
0.023154.
Ans.
12284.
Ans.
85.174.
Ans.
1.1185.
Ans.
1.6035.
Ans.
3.558.
Ans.
0.26615.
Ans.
67.217.
Ans.
Am.
0.064507.
19.
'.'.
(41.73
249 )
Ans.
21. (0.1234)°.216.
22. (0.1234)2.16.
23. (0.1234)-0.216.
24. (0.1234)-2.16.
25. (1.234)-0.216.
26. (1.234)-2.16.
27. (1.234)21.6.
28. (0.004)°.564.
29. (0.004)-0.00564.
30. (5.67)5.67.
31. (6.0606)6.06).
32.
33.
34.
35.
36.
37 .
38.
39.
57.692 X \,/93.2764.
Y;61.0061 X 0.0077079.
{/0.00006568 -+ y-'6:OO0888444.
\,/0.0064392 -+ ..y -0.00965432.
6607 X 8 X 91
0.01002 X 303.033 X 8.71'
576.9 Y 0.98764 X 57.
98.439 X 39.846
8.72 X y72.56 X (3.2654)2.
(2.3849)3 ..y974.681
7°.63(10°.34- 0.2°.34).
t-'nn
40.
Ans.
Ans.
Ans.
-
Ans.
8.2796.
Ans.
5.8891.
Ans.
16.127.
A.ns.
22-04.2,
9) = 1.
110
log
100
-
(
X
).
50. 2 log x = 2 +
51. 3x+y = 6Y, and 2x = 2(3)Y+1.
52. (10)x(0.01)Y
53. log x +
+
3)
=
1.
103.14.
Ans.
38,435.
Ans.
Ans.
= 1, and (0.003)1Y= 1.
log (x
Ans.
Ans.
-4.9233.
Ans.
-1.078.
Ans.
1.8415.
Ans.
-6.5784.
Ans.
1.
46. (0.036)x = 36.
47. 7x - 3(7)1x - 18 = O.
48. (0.9)2x + 3(0.9)x - 10 = O.
+
for horsepower,
H
where
2s = a
+
b
+
c
and
Ans. 17.904.
= PLAN
33000;
,
find H when P
=
Ans. 140.25.
76.5, L = 2.25, A = 231.8, and N = 116.
59. Given W = 0.0033 X 1O-7n, find W when n = 75,000.
Ans. 0.000024749.
60. In finding the diameter of a wrought-iron shaft that will transmit
90 hp. when the number of revolutions is 100 per minute, using a factor of
safety of 8, it is required to find the diameter d from the formula:
d = 68.5
:{/
~
W =
62.
= 3.
(x
58. Usmg the formula
0.91.
0.0000067896.
+ log
b)(s - c),
- a)(s -
ys(s
90
100 X 50,~00'
Ans.
3.5904.
Wgl'
M = 4bd3B' when g = 980,
75, l = 50, b = 0.98178, d = 0.5680, and B = 0.01093.
Ans. 11.681 X 10".
.
360 Lmgl when L
Fmd the value of n from the formula n = 1f"2()r4
= 69.6,
'
()
10, g = 980, l = 28,
= 1.1955, and r = 0.317.
Ans. 0.57704 X 10".
If 'In = ar 1.16, find r when m = 2.263 and a = O.40S6.
Ans. 0.22864.
-r
61. Find the value of M from the formula
43. Evaluate 50(el + e-i), where e = 2.71828.
44. 2088~ X 25(14,400~ - 2088').
Solve the following equations:
49. log x
57. Evaluate
a = 4.2763, b = 9.9264, and c = 8.4399.
.
0.84398.
-2.2842.
41. -4500 X 5!v-tU.
Ans. 5000.
Suggestion.-First substitute 8 for v and then substitute 4; finally, sub.
tract the second result from the first.
Ans.
27,806.
42. 11,52g.4~ 41.4\4-0.41 - 8-0.41).
45. (0.8)x
19
TABLES
~
1,818,700.
Ans.
OF
54. 2" + 5y = 1.64, and 42" = 3.9. Ans. x = 0.49087, Y = -0.90060.
55. (2.16)"(1.06)y = 0.12, and (3.8)"(4.9)Y = 2.7.
Ans. x = -2.9905, Y = 3.1372.
s(s - b)(s - c)
56. Evaluate
, where 2s = a + b + c and a = 47.236,
s - a
Ans. 31.750.
b = 82.798, and c = 75.643.
0.43146.
Ans.
3.6498.
Ans.
0.6364.
Ans.
0.010896.
Ans.
1.5713.
Ans.
91.78.
Ans.
0.95558.
Ans.
0.63497.
Ans.
93.866.
Ans.
0.04442.
Ans.
1.0316.
Ans.
18.741.
Ans.
55.209.
Ans.
110.29.
20. {/647.647.
AND EXPLANATION
LOGARITHMS
TRIGONOMETRY
PLANE
18
x
=
-1.691,
Ans.
10.
Y = -2.691.
x = 0.13312,
Y = 0.06656.
Ans.
2.
m =
n
63.
64. Given p = po(
1.41.
'Y ~ 1
)-r -1, find the value of p in terms of po if
'Y
=
Ans. 0.5266po.
65. If an indebtedness
is paid in instalments,
the payments being equal
and each including the interest to the date of the instalment, then the number of instalments necessary to pay the debt is given by the formula:
n
=
log p -log
log (1
(p-Pr),
+ r)
where P equals the total indebtedness,
p = the amount of one instalment,
Find the
and r = the rate per cent for the period between instalments.
number of instalments necessary to pay an indebtedness
of $1500 if the interest is 8 per cent per annum and the instalments
are $15 a month.
Ans. 164.5 nearly.
66. Using the formula of Exercise 65, find the number of instalments
if
the indebtedness
is $1800, the interest 5 per cent per annum, and the
instalments
$5 per week.
t
20
LOGARITHMS
PLANE TRIGONOMETRY
TABLE II
Using these values for
ing:
(6)
(6)
~ and M,
M and
x = 10gbN; then bx = N.
.'. loga bx = loga Nj or x loga b = loga N.
1
.'. x= l oga b loga N.
(3) and (4) become the follow-
logeN = 2.3026 lOg10N.
lOg10N = 0.43429 log eN.
24. Use of Table II.-In
~
to facilitate
Table II are arranged multiples of
changing natural
logarithms
to common
logarithms and vice versa.
Example I.-Find
the Napierian logarithm of 225, its common
logarithm being 2.35218.
1
By (3) (Art. 23) log. 225 =
loglO 225
M
1
= M X 2.35218.
~
(1)
21
lOg~10 = loglO e is usually represented by M.
1
1
Hence, by (2), = '
M
1oglo e
But
loglo e = loglo 2.71828 . . . = 0.43429448.
1
.'. M = 0.43429448 and M = 2.30258509.
~
But
OF TABLES
The modulus
23. Changing systems of logarithms.-In
what precedes, the
computations have been made with logarithms to the base 10.
It is often necessary to make computations when the logarithms
used are the natural, or N apierian, logarithms, in which the base
is e = 2.71828 . . '.
It will now be shown how to find the
logarithm of a number to the base e from the table of logarithms
to the base 10, and vice versa, by the help of Table II, page 56.
For the sake of generality, the relation between the logarithms
will be shown for any two bases.
THEoREM.--Given the logarithm of a number N to the base a;
then the logarithm of N to the base b is given by the relation:
1
10gb N =
loga N.
l oga b
Proof.-Let
AND EXPLANATION
x = 10gbN.
1
.' .10gb N= l oga b loga N.
~
1
M
system of which the base is b with reference to the system of which
the base is a.
If a is put for N in formula (1),
1
10gb a = loga a.
Ioga b
1
(2)
or 10gba loga b = 1.
.'. 10gba =
loga b'
It follows from (1) that the modulus of the natural system with
~
~
'.
~
reference to the common system is
That is,
(3)
(4) and
= 5.295945714
X 0.052
=
0.1197344248
X 0.00018 = 0.0004144653
~ .X 2.35218
= 5.4160946041
. . log. 225 = 5.41609.
Example 2.-Find
the common logarithm of 762, its natural
logarithm being 6.63595.
By (4) (Art. 23) loglo 762 = M log. 762 = M X 6.63594
l~'
oglo e and the modulus of the
common system with reference to the natural system is
X 2.3
M X 6.6
= 2.866343581
M X 0.035
= 0.01520030687
M X 0.00094 = 0.0004082368
log~ 10'
1
log. N = 1oglo e loglo N,
1
log. N.
loglO N =
log. 10
. . . M X 6.63594 = 2.88195212467
. . . loglo 762 = 2.88195
-
&
l
LOGARITHMS
PLANE TRIGONOMETRY
22
Example 3.-Find
10go.30.00107.
1
By Art. 23, 10go.30.00107 =
X 10glO0.00107
1OglO0 .3
1
= 1.47712 X 3.02938
-2.97062
= -0.52288 = 5.6814.
26. Evaluate![
Find the following logarithms:
1. log, 426.
2. log, 1076.
3. log, 0.0763.
4. log, 1.467.
6. log, 0.01352.
6. log, 0.002457.
7. log, 5.128.
8. log, 11'.
9. logo.3 3.16.
10. log2 9.23.
11.
12.
13.
14.
16.
Am. 6.05444.
Ans. 6.98101.
Ans. -2.57309.
Ans. 0.38322.
Ans. -4.3036.
Ans. -6.0088.
Ans. 1.6347.
Ans. 1.1447.
Ans. 1.0444.
Ans. 3.2064.
Ans. -0.0057949.
Ans. 2.17938.
Ans. 1057.
Ans. 0.031131.
Ans. 0.0857.
logo.3 1.007.
logloo 22,843.
Find x if log, x = 6.96319.
Find x if log, x = -3.46954.
Find x if log, x = -2.45673.
+
x2
+ 2 log,
(x
23
OF TABLES
+ v'4+X'> J:'
Ans. 5.916.
i[ xv
X2 - 16 - 16 log, (x + V x2 - 16) J:' Ans. 12.88.
TABLE III
25. On pages 58 to 65 are arranged the logarithms of sines and
tangents of angles from 3 to 7° for every 10"; and the logarithms
of cosines and cotangents of angles from 83 to 87° for every
10". The method of using these pages will follow from the
explanation for the remaining pages of the table (see Art. 28).
On pages 66 to 110 are arranged the logarithms, to five decimal places, of the trigonometric sines, cosines, tangents, and
cotangents, of angles from 0 to 90°, for each minute.
The logarithms in the columns headed log sin, log cos, or log
tan are increased by 10 so as to avoid writing negative characteristics. Those in the column headed log cot are printed without
this increase.
The minus sign is printed over the final 5 in the
logarithms, as explained in Art. 10.
The columns marked d give the tabular differences for the log sin
and log cos columns. The column marked c.d. (common difference)
gives the tabular differences for both log tan and log cot columns.
The marginal tables, marked Prop. Parts, give -r,1o,
,,\, . . . 11f'
it, H-, . . . H of the tabular differences, and are arranged for
27. Evaluate
EXERCISES
!xv' 4
AND EXPLANATION
snnuar
18. Evaluate 6 log, x]:.
-t
19. Evaluate -6log,( -x) ]
.
-4
20. Evaluate 6 + log, 4 - log, 2.
21. Evaluate -llS(log, 10 - log, 4).
log, 10.1 - log, 10 .
22 E va 1uate
0.0001
(
V
0)'
16.636.
Ans.
12.477.
Ans. 6.6931.
Ans. 0.02291.
.
23. Given R = 106. b .
log,
Ans.
A ns. 99.472.
where t = 120, V 0 = 123, V = 115.8,
V
Ans. 2.426 X 10'0.
and C = 0.082; find R.
24. The work W done by a volume of gas, expanding at a constant temperature from volume Vo to volume V" is given by the formula:
W
=
poVo log,
(iJ
Find the value of W if po = 87.5, Vo = 246, and V, = 472.
Ans. 14,026.
26. Given q = qoe"'; find k if q = !qo when t = 1800; then find q in terms
log,2
Ans. k = -1800' q = 0.8248qo.
of qo when t = 500.
~
and csc e = -J:- the logarithms of the
cos e
sm e'
secant and cosecant of an angle are the cologarithms (arithmetical
complements) of those of the cosine and sine respectively.
26. To find the logarithmic function of an acute angle.(1) When the angle is given in degrees and minutes. If the angle
is less than 45°, the degrees are found at the head of the page,
the minutes at the left, and the functions are taken as named
at the tops of the columns. If the angle is between 45 and 90°,
the degrees are found at the foot of the page, the minutes at the
right, and the functions are taken as named at the bottoms of
the columns. The functions are found in the same line with the
minutes.
The following should be located in the table.
log sin 17° 27' = 9.47694 - 10. log sin 68° 23' = 9.96833 - 10.
Since sec e =
log cos 29° 36' = 9.93927
log tan 10° 16' = 9.25799
-
10. log cos 76° 14' = 9.37652
10. log tan 86° 14' = 1.18154.
log cot 9° 46' = 0.76414. log cot 56° 43' = 9.81721 - 10.
-
10.
l
24
PLANE TRIGONOMETRY
LOGARITHMS
(2) When the angle contains seconds. Here the function is
found for the degrees and minutes and an interpolation made for
the seconds similar to the interpolations in Table 1. The tabular
d;ff~rence is multiplied by the number of seconds and divided by
6U. This product may be taken from the Prop. Parts tables.
Since the sine and tangent increase as the angle increases from 0
to 90°, the correction for the seconds is added; but, since the
cosine and cotangent decrease as the angle increases from 0 to
90°, the correction for the seconds is subtracted.
Example I.-Find
log sin 51° 26' 23".
Hence, if the increase in
function by 8 is x, then
14:8
.'. 9.81659 - 10
.'. e
for 23" = 10 X
. . . log
it =
OF TABLES
25
the angle necessary to increase the
= 60" :x". .'. x" = 34".
= log sin 40° 57' 34".
= 40° 57' 34".
Another angle having the same function is in the second
quadrant and is 180° - 40° 57' 34" = 139° 2' 26".
Example 2.-Find
e if log cos e = 9.23764.
Since the cosine decreases as the angle increases, locate in the
table the nearest log cos but larger than the one given.
9.23823 - 10 = log cos 80° 2'.
log sin 51° 26' = 9.89.314 - 10.
Correction
AND EXPLANATION
4
Tabular difference is 71.
Difference between the function given and the one found is 59.
sin 51° 26' 23" = 9.89318 - 10.
41 X 60" = 50".
Example 2.-Find log cos 27° 49' 37".
log cos 27° 49' = 9.94667
-
10.
Correction for 37" = 7 X it =
4
.
. . log cos 27° 49' 37" = 9.94663
-
10.
. '. IJ = 80° 2' 50".
Another angle having the same function is in the fourth quadrant
and is
RULE.-Find the function corresponding to the given degrees and
minutes. Multiply the tabular difference by the number of secQnds
considered as sixtieths. When .finding the sine, or tangent, add this
prod71rt to the fU/7ctio/7 corresponding to the degrees und minute8;
ut when jindmg the cosme, or cotangent, subtract this product.
(3) When the angle has decimal of minute. Here the only
difference in procedure from that given in (2) is that, in interpolating, the tabular difference is multiplied by the decimal of a
minute given. It is evident that the Prop. Parts tables cannot be
used for this.
27. To find the angle corresponding to a given logarithmic function.-(I)
When the function can be found in the table,
locate the function and read the angle in degrees and minutes at
the head and left, or at the foot and right, of the page, as the case
may be.
(2) When the function cannot befound in the table, the method of
procedure can best be shown by examples.
Example I.-Find the angle e if log sin e = 9.81659 - 10.
Nearest log sin but less from table, 9.81651 - 10 = log sin 40° 57'.
Tabular difference for a difference of l' in angle is 14.
Difference between given function and function found is 8.
}
TllH_1..,..t}(,
to 11 which is 8.3, the difference for 20". Then subtract 8.3
from 11 leaving 2.7, the difference for 6".
'. 9.98773 = log tan 44° 11' 26".
. '. e = 44° 11' 26" and 224° 11' 26".
Of course, the interpolation should be done mentally when
possible.
The method of procedure may be stated in the following rules:
RULE I.-For a logarithmic sine or tangent: (1) find the degrees
and minutes corresponding to the function next less than the given
function; (2) find the difference between the given function and the
one next less; (3) find the fractional part of 60" that this difference
is of the tabular difference. The required angle is the degrees and
minutes corresponding to the function found in the table together
with the seconds found.
RULE n.-For
a logarithmic cosine or cotangent: (1) find the
degrees and minutes corresponding to the function next greater than
.
A
26
LOGARITHMS
PLANE TRIGONOMETRY
the given function;
(2) find the difference between the given function
and the one next greater; (3) find the fractional part of 60" that this
difference is of the tabular difference.
The required angle is the
degrees and minutes corresponding
to the function found in the
table together with the seconds found.
28. Angles near 0 and 90°.-In what precedes, it has been
assumed that the variation in the angle is proportional to the
variation in the function.
In angles near 0°, this is not very
accurate with the sine and tangent; and near 90°, it is not very
accurate with the cosine and cotangent.
Table III (pages 58 to 65) gives the functions for every 10"
between 3 and 7° and 83 and 87°. This makes the interpolations
more nearly accurate for these angles. For the angles less than
3° and greater than 87°, the Sand T scheme is convenient.
29. Functions by means of Sand T.-The quantities Sand T
which are used are defined by the equations:
and
sIn a
.
or S = log sm a - log a,
S = log -,
a
tan a
or T == log tan a -log a,
T = log~~,
a
OF TABLES
27
EXAMPLES
1. Find log sin 0° 47' 19".
47' 19" = 2839"
log 2839 = 3.45317
S=4.68556-1O
3. Find log cot 0° 57' 49".
57' 49" = 3469"
!\pllog 3469 = 6.45980 -10
cpl T=5.31438
.', log sin 0° 47' 19" =8 .13873-10
. . . log cot 0° 57' 49" = 1.77418
4. Find
2. Find log tan 1° 27' 14".
1°27' 14"=5234"
log 5234 = 3 .71883
T =4.68567-10
I
a if log sin a = 7 .85387 -10
log sin a = 7.85387 -10
cplS=5.31443
. . , log a" = 3 .16830
a" = 1473.3"
. . , log tan 1° 27' 14" = 8 .40450 -10
5. Find log cos 89° 27' 32".
90° - 89° 27' 32" = 0° 32' 28"
= 1948"
log 1948 = 3 .28959
S =4.68557 -10
. . . log cos 89° 27'
where a is the number of seconds in the angle.
For convenience, the values of S, T, and a for angles from 0°
to 3° 4' are arranged at the bottom of pages 36 to 55.
On pages 66 to 68 are columns headed cpl Sand cpl T. There
give the arithmetical complements of the values of Sand T.
From the above are derived the following:
AND EXPLANATION
32" =7.97516
-10
,',
7. Find log tan 89° 47' 33.82".
90°-89° 47' 33.82"=12'
26.18".
=746.18"
colog 746.18 =7 .12715-10
cpl T=5.31442
I .'.
-log tan 89° 47' 33.82"
8. Find
6. Find log cot 88° 49' 51",
90°-88° 49' 51"=1° 10' 9"
= 4209"
log 4209 =3 .62418
T =4.68563-10
. . . log cot 88°49' 51" = 8.30981-10
a =0° 24' 33.3"
=2.44157.
a if log cot a = 7.86432 -10
log cot a = 7.86432-10
cpl T = 5.31442
.'.log
(900-a)"=3.17874
(90° -a)" = 1509.2"
90° - a = 0° 25' 9.2"
. . . a =89° 34' 50.8"
FORMULAS FOR THE USE OF SAND T
(1) For angles near 0°.
log sin a = log a" + S.
log tan a = log a" + T.
log cot a = epIlog a" + cpl T
= epIlog tan a.
log a" = log sin a
= log tan a + cpl T
= epIlog cot a + cpl T.
(2) For angles near 90°.
log cos a = log (90° - a)" + S, log (90° -a)"
log cot a = log (90° - a)" +T.
log tan a = epIlog (90° - a)"
+cpl T
epIlog
cot a.
=
+ cpl S
= log cos a
+ cpl S
= log cot a + cpl T
= cpl log tan a
cpl T.
+
In Example 4, to find cpl S, locate log sin a on page 66, and read cpl S in
the adjoining column. In Example 8, to find cpl T, locate log cot a on page
66.
30. Functions of angles greater than 90°.-In
trigonometry
there is a rule which says: To find the function of an angle
greater than 90°, express the angle as a multiple of 90° plus an
acute angle. If this multiple is even, take the same function of
the acute angle as the one required; and, if the multiple is odd,
take the cofunction of the acute angle. In either case prefix
the sign determined by the quadrant the original angle is in.
Thus,
sin 562° = sin (6 X 90° + 22°) = -sin 22°
tan 1042° = tan (11 X 90° + 52°) = -cot 52°.
~
28
PLANE TRIGONOMETRY
LOGARITHMS
As a further convenience in finding the logarithmic functions of
angles greater than 90°, there are arranged at the top and bottom
of each page of Table III other angles. If a is the acute angle of
the page, then 180°
+
a is printed
functions as a; while 90°
+ a and
+ a are
(122.87)! X 0.00008721
29. Find x if x =
tan 180 19' 20"
EXERCISES
10 14' 27" = 8.33566.
8.32572.
= 9.43538. 10. log cot 89012' 18" = 8.14227.
= 9.85491. 11. log cas 2160 14' 33" = 9 90662n.
= 0.98972. 12. log sin 138048' 6" = 9.81867.
= 0.79889. 13. log tan 3250 17' 29" = 9.84052n
= 7.93765. 14. log cot 227028' 3" = 9.96253.
.Fmd the values of 0 less than 3600 in the following:
15. log sin 0
=
=
0.49632.
21. log sin 0
=
22. log sin
=
7.99892.
9.98762n.
9.89263n.
18. log cot
0
19. log tan 0 = 0.49936.
20. log cas 0 = 8.32967.
0
23. log cas 0 =
8. log tan
= 9.92940. 9. log cas 88047' 13" =
9.28762.
16. log cas 0 = 9.87642.
17. log tan 0 = 9.47632.
Ans.
110 10' 53" and 1680 49' 7".
Ans. 410 12' 22" and 3180 47' 38".
Ans. 16040' 13" and 196040' 13".
Ans.
17041'
0.0013093,
31. In this table (pages 112 to 134) are arranged the natural
trigonometric sine, cosine, tangent, and cotangent of angles from
0 to 90° for each minute.
The values are given correct to five
figures.
The arrangement of and the method of using the table are
practically the same as for Table III, except that there are no
differences given and no table of proportional parts.
32. To find the natural function of an angle.-(I)
When the
angle is given in degreesand minutes: If the angle is less than 45°,
the degrees are found at the head of the page, the minutes at the
left, and the functions are taken as named at the tops of the
columns. If the angle is between 45 and 90°, the degrees are
found at the foot of the page, the minutes at the right, and the
functions are taken as named at the bottoms of the columns.
The functions are found in the same line with the minutes.
When the angle is greater than 90°, first express as a function
of an acute angle, as ill Art.
Thus,
The small letter n is placed after the function to indicate that
the natural function is negative.
Of course, the logarithm cannot
take account of this.
= 9.94469.
Ans.
859.44.
TABLE IV
log cos 128° = log sin 38°n.
log tan 218° = log tan 38°.
log sin 308° = log cos 38°n.
Verify the following.
1. log sin 610 41' 31"
2. log cas 310 47' 27"
3. log tan 150 14' 36"
4. log sin 450 43' 28"
5. log cot 5050' 47"
6. log tan 800 58' 17"
7. log sin 0029' 47"
Ans.
29
printed in black
type, and for the functions of these angles one must take the cofunction of a. In either case proper regard must be paid to the
algebraic sign.
Thus,
OF TABLES
1906
28. Find x if x =
cot 240 16' 19'"
in light type and has the same
270°
AND EXPLANATION
18" and 197041'
sin 27° 13'
cos 36° 42'
tan 11° 17'
cot 21° 43'
sin 228° 13'
18".
Ans. 72025' 38" and 2520 25' 38".
Ans. 880 46' 33" and 271 0 13' 27".
Am.
00 34' 17.5" and 1790 25' 42.5".
Ans. 256023' and 2830 37'.
Ans. 141020' 54" and 2180 39' 6".
0.45736.
sin 62° 18'
0.80178.
cos 83° 47'
0.19952.
tan 75° 14'
2.5108.
cot 56° 28'
-sin 48° 13' = -0.74567.
=
=
=
=
0.88539.
0.10829.
3.7938.
0.66272.
(2) When the angle contains seconds: Here the function is found
for the degrees and minutes as in (1) and an interpolation is
made for the seconds similar to the interpolation of Table I.
The tabular difference is multiplied by the number of seconds and
divided by 60. Since the sine and the tangent increase as the
angle increases from 0 to 90°, the correction for the seconds is
added; but, since the cosine and the cotangent decrease as the
angle increases from 0 to 90°, the correction for the seconds is
subtracted.
24. log tan 0 = 0.96236n.
Ans. 96013' 25" and 2760 13' 25".
3.26 tan 1980 13' cas 130 17'.
25 . t an 0 =
4.76 sin 280 16'
Ans. 24051' 15" and 2040 51' 15".
.
26 . Sill 0 -- 17 sin 2830 19' tan 470 16'.
39.2 cas 1830 6'
Ans. 270 13' 26" and 152046'
27. cas 20 = tan (-27120 15' 40") see 30500 40' .
tan 15220 46' 3D" csc 18980 17'
=
=
=
=
=
34".
Ans. 45047' 30", 134012' 30", 225047' 3D",and 314012' :to"
A
30
PLANE
Example I.-Find
LOGARITHMS
TRIGONOMETRY
for 16" = 26 X
. . . sin
Example 2.-Find
H =
7
27° 41' 16" = 0.46465
cot 65° 22' 36".
cot 65° 22' = 0.45854
21
Correction for 36" = 35 X H =
. . . cot 65° 22' 36" = 0.45833
Example
V.1
OJU
l/J..J.v
.1VVlI
31
the value of angle (j if tan (j
3.-Find
= -1.2783.
Since tan (j is negative, (j must lie in the second and the fourth
quadrants.
First find the angle (j' in the first quadrant that has its tangent
RULE.-Find
the function corresponding to the given degrees
and minutes.
Multiply the tabular difference by the number of
seconds considered as sixtieths. When finding the sine, or tangent,
add this product to the function corresponding to the degrees and
minutes; but, when finding the cosine, or cotangent, subtract this
product.
(3) When the angle has decimal of minute: Here the only difference in procedure from that given in (2) is that, in interpolating,
the tabular difference is multiplied by the decimal of a minute
gIven.
33. To find the angle corresponding to a given natural function.
(1) When the function can be found in the table, locate the function and read the angle in degrees and minutes at the head and
ellJ,
OF TABLES
Tabular difference for difference of I' in angle is 27.
Difference between given function and function found is 18.
The increase in the angle necessary to decrease the function by
18 is -H X 60" = 40".
.'.0.32346 = cos 71° 7' 40".
. . . (j = 71° 7' 40".
Another angle having the same cosine is in the fourth quadrant
and is 360° - 71° 7' 40" = 288° 52' 20".
sin 27° 41' 16".
sin 27° 41' = 0.46458
Correction
EXPLANATION
AND
to tan (j.
equal numerically
Then
That
is, find (j' if tan (j'
= 1.2783.
(j' = 51° 57' 53".
(j
= 180°
(j
= 360°
or
-
(j'
= 128° 2' 7",
-
(j'
= 308° 2' 7".
EXERCISES
Verify the following:
1. sin 270 22' 41" = 0.45986.
8. tan 1560 42' 13" = -0.43059.
2. cos 360 14' 16" = 0.80657.
9. sin 2200 35' 30" = -0.65066.
3. tan 410 19' 26" = 0.87926.
10. cot 295017' 14" = -0.47242.
4. cot 130 14' 52" = 4.2475.
5. cos 720 28' 14" = 0.30119.
6. tan 830 40' 30" = 9.0218.
11
11. cos 3140 14.6' = 0.69771.
12. sin 1260 23.7' = 0.80494.
13. Rin 342043.2' = -0.29704.
£..4IJ.J.\.A. J..I.6J..I.~,
Find the values of 0 less than 3600 in the following:
15. sin 0 = 0.49367.
Ans.
290 34' 55" and 1500 25' 5".
16. sin 0 = 0.82764.
Ans.
55051' 26" and 1240 8' 34".
17. cos 0 = 0.89672.
Ans.
260 16' 10" and 3330 43' 50".
18. cos 0 = 0.22724.
Ans.
760 51' 56" and 2830 8' 4".
(2) When the function cannot be found in the table, the method
involves interpolation and can best be shown by examples.
Example I.-Find
the values of angle (j if sin (j = 0.53862.
Nearest sine but less from table, 0.53853 = sin 32° 35'. Tabular difference for difference of I' in angle is 24. Difference
between given function and function found is 9.
The increase in the angle necessary to increase the function
by 9 is -h X 60" = 22t" or 23".
. . . 0.53862 = sin 32° 35' 23".
.'. (j = 32° 35' 23".
Another angle having the same sine is in the second quadrant
and is 180° - 32° 35' 23" = 147° 24' 37".
(j
Example 2.-Find the value of angle (j if cos = 0.32346.
Since the cosine decreases as the angle increases, locate in the
table the nearest cosine but greater than the one given. This is
0.32364 = cos 71° 7'.
19. tan
0
=
2.4379.
Ans.
20.
21.
22.
23.
24.
25.
26.
0
0
0
0
0
0
0
=
=
=
=
=
=
=
0.87623.
1.8923.
0.43729.
-0.89723.
-0.42936.
-0.92834.
-2.4376.
Ans.
Ans.
Ans.
Ans.
Ans.
Ans.
Ans.
tan
cot
cot
sin
cos
tan
cot
670 41' 49"
and
2470 41' 49".
410 13' 33" and 2210 13'
270 51' 17" and 2070 51'
660 22' 51" and 2460 22'
243047' 46" and 2960 12'
115025' 37" and 2440 34'
1370 7' 41" and 3170 7'
157041' 40" and 3370 41'
33".
17".
51".
14".
23".
41".
40".
TABLE V
34. This table (page 135) can be used to change an angle
expressed in degrees to radians, or vice versa. It may also be used
for finding the arc length in a circle when the angle at the center
is given, or vice versa.
&.
32
Example
PLANE
L-Express
LOGARITHMS
TRIGONOMETRY
y
The accuracy of the last figure in the sum cannot be relied upon.
Example 2.-Express
3.6678437 radians in degrees, minutes,
and seconds.
Given, 3.6678437
Next less in table, 3.1415927 = 1800
Difference, 0.5262510
N ext less, 0.5235988 = 300
Difference, 0.0026522
00 9'
Next less, 0.0026180 =
Difference, 0.0000342
000' 7"
Next less, 0.0000339 =
Difference, 0.0000003
.' . 3.6678437 radians = 2100 9' 7".
1.6
1
0.6
, I.
0
."
I,
, ..
I ",.1..
10
L
20
16
Y=
x
L
26
log X
FIG.
EXERCISES
1.
.26240
.26215
4. 3.96423radians = 2270 x' 2~ .
. 5. 1.49367radians = 85034'
52".
6. 0.0236784radian = 10 21' 24".
.26190
35. Errors ofinterpolation.-In
the process of interpolation
in logarithms, values are inserted as if the change in the logarithm
between the two nearest tabular values was directly proportional
to the change in the number.
This would mean that the graph
of the equation y = log x for this interval is a straight line.
If values of x and y = log x are plotted in the usual manner in
rectangular coordinates, the graph of y = log x is as shown in
Fig. 1, where the unit on the y-axis is 10 times as large as the unit
on the x-axis.
The values of the logarithms of numbers can be read from this
curve, but not to a very high degree of accuracy.
The values of
x and y given in the table fall so close together on this curve that
the interpolating cannot be shown. Suppose, for example, that
log 1.7854 is required.
Take the portion of the curve near
x = 1.7854 and magnify it in the ratio of 1 to 20,000 on the x-axis
----------------------------------
6
y
Verify the following:
1. 216044' 44" = 3.78292 radians.
2. 47023' 58" = 0.82728 radian.
-..
33
OF TABLES
and 1 to 1000 on the y-axis; the resulting curve is shown in Fig. 2.
Referring to this figure, when x = 1.785, y = 0.25164; and when
x = 1.786, Y = 0.25188. These give, respectively, the two points
Sand T on the curve. When x = 1.7854, y = log 1.7854 has the
value shown by the point P on the curve; but, by interpolation,
143027' 36" in radians.
1430 = 2.4958208
27' = 0.0078540
36" = 0.0001745
. '. 143027' 36" = 2.5038493 radians.
--------
AND EXPLANATION
---
.26166
.00048
.000336
.26142Olf.784
x
1.785
1.786
1.787
Y=logX
FIG. 2.
the value of log 1.785i
= 0.251736
and is shown by the point
Q.
Therefore, the interpolation gives an error equal to QP.
By using a higher-place table of logarithms, the value of log
1.7854 = 0.2517355. This shows that the error is such that the
logarithm is not affected in the fifth decimal place.
A similar discussion could be given for interpolating in trigonometric functions.
--
I
l
TABLE I
COMMON
LOGARITHMS
OF NUMBERS
From 1 to 10,000 to five places.
From 10,000 to 11,000 to seven places.
(For explanations, see pages 7 to 12.)
Also values of S. and T. from 0° to 3°4'.
(For explanations, see page 26.)
.
I
l
TABLE I
N. !L.
100
101
102
103
104
106
106
107
108
109
110
III
112
113
114
116
116
117
118
119
120
121
122
123
124
126
126
127
128
129
130
131
132
133
TABLE I
100.150
()
:r
2
3
00 000 043 087
432 475 5\8
860 903 945
01 284
326 368
745 787
703
02 119 160 202
531 572 612
938 979 *019
03 342 383 423
743 782 822
04 139 179 218
532 571
610
922 961 999
05 308 346 385
690 729
767
06 070
108 145
446 483
521
819 856 893
07 188 225 262
555 591
628
918 954 990
08 279 314 350
636 672
707
99\ *026 *061
09 342 377 412
691 726 760
10 037 072
106
380 415 449
721 755 789
11 059 093
126
394 428 461
727 760 793
130
561
988
410
828
243
653
*060
463
862
258
650
*038
423
805
183
558
930
298
664
*027
386
743
*096
447
795
140
483
823
160
494
826
4
173
604
*030
452
870
284
694
*100
503
902
297
689
*077
461
843
221
595
967
335
700
*063
422
778
132
* 482
830
175
517
857
193
528
860
5
6
I
7
8
1
2171
647
*072
494
912
325
735
*141
543
941
336
727
*115
500
881
258
633
*004
372
737
*099
458
814
167
* 517
864
209
551
890
227
561
893
260
689
* 115
536
953
366
776
*18\
583
981
376
766
*154
538
918
296
670
*041
408
773
*135
493
849
*202
552
899
243
585
924
261
594
926
303
732
157
* 578
995
407
816
*222
623
*021
415
805
*192
576
956
333
707
*078
445
809
*171
529
884
*237
587
934
278
619
958
294
628
959
346
775
199
* 620
*036
449
857
*262
663
*060
454
844
*231
614
994
371
744
115
* 482
846
*207
565
920
*272
621
968
312
653
992
327
661
992
9
I
389
817
*242
662
*078
490
898
*302
703
*100
493
883
*269
652
*032
408
781
* 151
518
882
*243
600
9551
*307
656
*003
346
687
*025
361
694
*024
12 057
090
123
\56
189
222
254
287
320
3851
4181
4501
4831
516
548
58!
613
6461 *678
N.
Prop. Parts
44
4.4
8.8
13.2
17.6
22.0
26.4
30.8
35.2
39.6
41
4.1
8.2
12.3
16.4
20.5
24.6
28.7
32.8
36.9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
43
4.3
8.6
12.9
17.2
21.5
25.8
30.1
34.4
38.7
42
4.2
8.4
12.6
16.8
21.0
25.2
29.4
33.6
37.8
40
4.0
8.0
12.0
16.0
20.0
24.0
28.0
32.0
36.0
39
3.9
7.8
11.7
15.6
19.5
23.4
27.3
31.2
35.1
38
37
3.83.73.6
7.6 7.4
11.4 11.1
15.2 14.8
19.0 18.5
22.8 22.2
26.6 25.9
30.4 29.6
34.2 33.3'
35
34
36
3.5
352 1
3.4
6.8
~16'~
4 I 14.0
5 17.5
6 21.0
7 24.5
8 28.0
9 31.5
32
I 3.2
2 6.4
3 9.6
4 12.8
5 16.0
6 19.2
7 22.4
8 25.6
9 28.8
7.2
10.8
14.4
18.0
21. 6
25.2
28.8
32.4
33
3.3
6.6
I
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
13 033; 0661 098: 130; 162 194! 2261 2581 2901 322
354 386 418 450 481 513 545 577 609 640
672 704
735 767 799 830 862 893 925 956
988 *019 *051 *082 *114 *145 *176 *208 *239 *270
14 301 333 364 395 426
457 489 520 551 582
613 644 675 706 737 768
799 829 860 891
922 953
983 *014 *045 *076 *106 *137 *168 *198
15 229 259
290 320 351 381 412 442
473 503
534 564
594 625 655 685 715 746
776 806
836
866 897 927 957 987 *017 *047 *077 *107
16 137 167
197 227 256 286
316 346
376 406
435
465 495 524 554 584 613 643 673 702
732 761 791 820 850 879 909
938 967 997
17 026 056 085
114 143 173 202 231 260 289
319 348 377 406 435 464 493
522 551 580
6091 638
667 696 725
754 782
811 840 869
N. I L.
0
I
2
I
3
5
6
7
8
9
Prop.
13.6
17.0
20.4
23.8
27.2
30.6
31
3.1
6.2
9.3
12.4
15.5
18.6
21.7
24.8
27.9
13.2
'-!
16.5
.19.8
23.1
26.4 .
29.7
30
3.0
6.0
9.0
12.0
15.0
]8.0
21. 0
24.0
27.0
Parts
160
151
152
153
154
166
156
157
158
159
160
161
162
163
164
166
166
167
168
169
170
171
172
173
174
176
176
177
178
179
180
\81
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
\97
\98
199
200
L.
150.200
0
17 609
898
18 184
469
752
19 033
312
590
866
20 140
412
683
952
21 219
484
748
22 011
272
531
789
23 045
300
553
805
24 055
304
551
797
25 042
285
527
768
26 007
245
482
717:
951
27 184
416
646
875
28 103
330
556
780
29 003
. 226
447
667
885
30 103
I
2
6381
926
213
498
780
061
340
618
893
167
439
710
978
245
511
775
037
298
557
814
070
325
578
830
080
329
576
822
066
310
551
792
031
269
505
7411
975
207
439
669
898
126
353
578
803
026
248
469
688
907
125
~I
I
3
4
!
2
I
3
I
15~
0° 2'= 120" S. 4.68557 T. 4.68557
0 3 = 180
557
557
0 4 = 240
557
558
0 25 1500
557
558
0 26 =
557
558
= 1560
0 27 =
1620
557
558
0
0
0
l' = 60" S. 4.68557 T. 4.685571 0° 19' = 1140" S.4.68557 T. 4.68 558
2 = 120
557
557 0 20 = 1200
557
558
3 = 180
557
557 0 21 = 1260
557
558
558
557
0 22 = 1320
558
0 16 = 960
557
557
0 23 = 1380
558/
558
557
0 17 = 1020
557
558 0 24 = 1440
558
558 0 25 = 1500
557
0 18 = 1080
557
36
I
5
667 696 725 754:
955 984 *013 *041
241 270 298 327
526 554 583 611
808 837 865 893
089
117 145 173
368
396 424 451
645 673 700 728
921 948 976 *003
194 222 249 276
466 493 520 548
737 763 790 817
*005 *032 *059 *085
272 299 325 352
537 564 590 617
801 827 854 880
063 089
115 141
324 350 376 401
583 608 634 660
840 866 891 917
096
121 147 172
350 376 401 426
603 629 654 679
855 880 905 930
105 ]30
155 180
353 378 403 428
601 625 650 674
846 871 895 920
091
115 139 164
334 358 382 406
575 600 624 648
816 840 864 888
055 079
102 126
293 316, 340 364
I
764: 7881 811 834!
998 *021 *045 *068
231 254 277 300
462 485 508 531
692 715 738 761
921 944 967 989
149 171 194 217
375 398 421 443
601 623 646 668
825 847 870 892
048 070 092
115
270 292 314 336
491 513 535 557
710 732 754 776
929 951 973 994
146 168 190 211
6
I
7
I
8
I
9
782 811
840
*070 *099
127
*
355 384 412
639 667 696
921 949
977
201 229 257
479 507 535
756 783 811
*030 *058 *085
303 330 358
575 602 629
844 871' 898
*112 *139 *165
378 405 431
643 669 696
906 932 958
167 194 220
427 453 479
686 712 737
943 968 994
198 223 249
452 477 502
704 729
754
955 980 *005
204 229 254
452 477
502
699 724 748
944 969 993
188 212 237
431 455 479
672 696 720
912 935 959
150 174 198
387 411 4351
869
*156
441
724
*005
285
562
838
112
* 385
656
925
*192
458
722
985
246
505
763
*019
274
528
779
*030
279
527
773
*018
261
503
744
983
221
458
858: 8811
*091 *114
323 346
554 577
784 807
*012 *035
240 262
466 488
691 713
914 937
137 159
358 380
579 601
798 820
*016 *038
233 2551
928
*161
393
623
852
*081
307
533
758
981
203
425
645
863
*081
298
Prop. Parts
I
1
2
3
4
5
6
7
8
9
29
2.9
5.8
8.7
11.6
14.5
17.4
20.3
23.2
26. I
I
2
3
4
5
6
7
8
9
27
2.7
5.4
8.1
10.8
13.5
16.2
18.9
21 .6
24.3
I
2
3
4
5
6
7
8
9
1
2
I
28
2.8
5.6
8.4
11.2
14.0
16.8
19.6
22.4
25.2
26
2.6
5.2
7.8
10.4
13.0
15.6
18.2
20.8
23.4
26
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
22.5
24
23
2.4
2.3
4.8
4.6
1
9051
*138
370
600
830
*058
285
511
735
959
181
403'
623
842
*060
276
41 9.6
5 12.0
6 14.4
7 16.8
8 19.2
9 21.6
22
2.2
1
4.4
2
3
6.6
4
8.8
5 11.0
6 13.2
7 15.4
8 17.6
9 19.8
9.2
11.5
13.8
16.1
18.4
20.7
21
2.1
4.2
6.3
8.4
10.5
12.6
14.7
16.8
18.9
Prop. Parts
4
0°
0
0
0
0
0
0
28'=
29 =
30 =
31 =
32 =
33 =
34 =
1680" S. 4.68557 T. 4.68558
1740
557
559
1800
557
559
1860
557
559
1920
557
559
1980
557
559
2040
557
559
37
.
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
0
I
30 103
320!
5~51
750,
963'
31 175
387
597
806
32 015
33
34
35
36
37
38
39
2221
428
634
838
041
244
445
646
846
044
242
439
635
830
025
218
411
603
793
984
173
361
549
736'
9221
107!
291
475
658
840
021
202
382
561
739
917
094
270
445
620
794
I
I
125
341
557
771
984
197
408
618
827
035
243
449
654
858
062
264
465
666
866
064
262
459
655
850
044
238
430
622
813
*003
192
3801
568
754!
9401
125!
310
493
676
858
039
220
399
578
757
934
111
287
463
637
811
2
I
146
363
578
792
*006
218
429
639
848
056
263
469
675
879
082
284
486
686
885
084
282
479
674
869
064
257
449
641
832
*021
211
399
586
773'
9591
144!
328
511
694
876
057
238
417
596
775
952
129
305
480
655
829
3
I
168
384
600
814
*027
239
450
660
869
077
284
490
695
899
102
304
506
706
905
104
301
498
694
889
083
276
468
660
851
*040
229
41~]
605
791'
9771
162'
346
530
712
894
075
256
435
614
792
970
146
322
498
672
846
~3 I
0°
3'=
I
4
190
406
621
835
"048
260
471
681
890
098
305
510
715
919
122
325
526
726
925
124
321
518
713
908
102
295
488
679
870
*059
248
436
624
810
996
181
365
548
731
912
093
274
453
632
810
987
164
340
515
690
863
4
I
5
I
233
449
664
878
*091
302
513
723
931
139
346
552
756
960
163
365
566
766
965
163
361
557
753
947
141
334
526
717
908
*097
286
4741
661
847'
*0,,1
218'
401
585
767
949
130
310
489
668
846
*023
199
375
550
533[
707 724
881 898
211
428
643
856
*069
281
492
702
911
118
325
53\
736
940
143
345
546
746
945
143
341
537
733
928
122
315
507
698
889
*078
267
455
642
829'
*0141
199'
383
566
749
931
112
292
471
650
828
*005
182
358
I
5 I
180" S. 4.68 557 T. 4.68557
4 = 240
5 = 300
557
557
558
558
I
0 33 =1980
0 34 =2040
0 35 =2100
557
557
557
559
559
559
I
0
0
6
6
0°
0
0
0
0
0
I
7
I
8
I
255
471
685
899
*112
323
534
744
952
160
366
572
777
980
183
385
586
786
985
183
380
577
772
967
160
353
545
736
927
*116
305
493
680
866
*0511
236'
420
603
785
967
148
328
507
686
863
*041
217
393
568
742
915
276
492
707
920
*133
345
555
765
973
181
387
593
797
*001
203
405
606
806
*005
203
400
596
792
986
180
372
564
755
946
*135
324
5111
698
884'
*0701
254'
438
621
803
985
166
346
525
703
881
*058
235
410
585
759
933
I
I
7
8
'9
I
298
514
728
942
*154
366
576
785
994
201
408
613
818
*021
224
425
626
826
*025
223
420
616
811
*005
199
392
583
774
965
*154
342
530
717
903
*088
273
457
639
822
*003
184
364
543
721
899
*076
252
428
602
777
950
9
Prop. Parts
1
2
3
4
5
6
7
8
9
22
2.2
4.4
6.6
8.8
11.0
13.2
15.4
17.6
19.8
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
I
2,1
4,2
1<
1<
6,3
8.4
10.5
12.6
14.7
16.8
18.9
20
2,0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
18.0
19
1.9
3.8
5.7
7.6
9.5
11.4
13.3
15.2
17.1
18
1
2
3
4
5
6
7
8
9
1
2
3
4!
5
6
7
8
9
21
1. 8
f
3.6
J:'f
7.2
9.0
10.8
12.6
14.4
16.2
17
1.7
3.4
5.1
6.8
8.5
10.2
11.9
13.6
r5.3
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
I 285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
250-300
0
2
3
4
39 794' 81\
829 846 863
967 985 *002 *019 *037
40 140 157 175 192 209
312329346361381
483 500 518 535 552
654 671 688 705
722
824 841 858 875 892
993 *010 *027 *044' *061
41 162 179 196 212 229
330347363380397414
497 514 531 547 564
664 681 697 7141 731
830 847 863
880' 896
996 *012 *029 *0451 *062
42 160 177 193 210 226
325 341 357 374 390
488 504, 521
537 553
651 6671 684
700 716
8\3
830 846 862 878
975 991 *008 *024 *040
43 136 152 169 185 201
297 313 329 345 361
457 473 489 505 521
616 632 648 664 680
775 79\
807 823 838
933 949 965 981 996
44 091
107 122 138 154
248 264 279 295 311
404 420 436 451 467
560 576 592
607 623
716 731 747 762 778
871 886 902 917 932
45 025 040, 056 071 086
179 194' 209 225 240
332 3471 362 378 1 393
48{
500' 5151 530: 545
637 652 667 6821 697
788 803 818 834 849
939 954 969 984 *000
46 090
105 120 135 150
240
255 270 285 300
389 404 419 434 449
538 553 568 583 598
68Z 7Q2 716 731 746
835' 850 864
879 894
982 9971 *012 *026 *041
47 129' 144i 159 173 188
276 290, 305 319 334
422 436 451 465 480
567 582 596 611 625
712 727 741 756 770
=
Prop. Parts
I
I
1
1
1
1
1
1
'
1
1
1
1
1
.
3
I
4
5
S. 4.68 557 T. 4.68 559
557
559
557
559
557
559
557
559
556
560
556
560
0 42 = 2520
7
8
1
1
1
'
0
Prop. Parts
9
1
0° 4'= 240" S. 4.68 557 T. 4.68558
0 5 = 300
557
558
0 41 = 2460
556
560
0 42 = 2520
556
560
0 43 = 2580
556
560
0 44 = 2640
556
560
36'=2160"
37 = 2220
38 = 2280
39 = 2340
40 = 2400
41 =2460
6
881 898 915' 933
*054 *071 *0881 *106
226
24~ 261
278,
398 415432449,466
569
586
603 620'
739
756 7731 7901
909
926 943 960
*078 *095 *111 *128
246
263 280 296
430447464481
581 597 614 631
747
764 780 797
913
929 946 963
*078. *095 *1111 *127
243
259 275 292
406
423 439 455
570
586 602 619
732
749 765 781
894
911 927 943
*056 *072 *0881 *104
217 233 249 . 265
377 393 409 425
537 553 569 584
696 712 727 743
854 870 886 902
*012 *028 *044 *059
170
185 201
217
326
342 358
373
483
498 514
529
638
654 669 685
793
809 824
840
948
963 979 994
102
117 133
148[
255
271 286 j 301:
'
408 I 423 t 439
454'
561! 576' 591! 606'
712
7281 743
7581
864
879 894 909
*015 *0301 *045 *060
165
180 195 210
315
330 345
359
464
479 494 509
613
627 642
657
761
776 790 805
909 923 938 953
*056 *070 *085 *100
202
217 232 2461
349
36}
378
392"
494 509 524 538
640 654 669 6831
784 799 813 828
'
1
I
L.
1
N.
TABLE I
N. I L.
200-250
TABLE I
950
*123
295
637
807
976
* 145
313
647
814
979
*144
308
472
635
797
959
*120
18
1.8
3.6
5.4
7 2
9: 0
10.8
12.6
14.4
16.2
17
1.7
3 4
5: 1
6.8
8.5
10.2
11.9
13 6
15.3
I
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
281 log e = 0.43429
441
600
759
917
*075
232
389
545
700
855
*010
163
317
469
621
773
924
*075
225
374
523
672
820
967
*114
261
407
553
698
842
16
1.6
3.2
4.8
6.4
8.0
9.6
11.2
12.8
14.4
15
1 1 1.5
2 I 3.0
4 5
6.0
4'
5
7.5
6
9.0
7 10.5
8 12.0
9 13 5
14
1.4
1
2
2.8
3
4 2
4
5:6
5
7.0
6
8.4
7
9.8
8 11.2
9 12.6
1
2
3
4
5
6
7
8
9
1
"
1
1
6
0°
0
0
0
0
0
17
I
45'=2700"
46 = 2760
47 = 2820
48 = 2880
49 =
2940
50 = 3000
8
~
Prop. Parts
S. 4.68 556 T 4.68 560
556
560
556
560
556
560
556
560
556
560
39
38
I
TABLE I
N. I L.
0
I
I.
300.350
516\71819
2
131
~
47 712 727 741 756 770 784
857 871 885 900 914 929
48 001 015 029 044 058 073
144 159 173 187 202 216
287 302 316 330 344 359
430 444 458 473 487 501
572 586 601 615 629 643
714 728 742 756 770 785
855 869 883 897 911 926
996 *010 *024 *038 *052 *066
49 136 150 164 178 192 206
276 290 304 318 332 346
415 429 443 457 471 485
554 568 582 596 610 624
693 707 721 734 748 762
831 845 859 872 886 900
969 982 996 *010 *024 *037
50 106 120 133 147 161 174
243 256 270 284 297 311
379 393 406 420 433 447
5 i 5 529 542 556 569 583
651 664 678 691 705 718
786 799 813 826 840 853
920 934 947 96! 974 987
51 055 068 081 095 108 121
188 202 215 228 242 255
322 335 348 362 375 388
455 468 481 495 508 521
587 I 601 614 627 640 654
720 . 733 746 759 772 786
851 865 878 891 904 917
983 996 *009 *022 *035 *048
52 114 127 140 153 166 179
241 257 270 284 ! 297 310
375
388 ' 401
414 i 427
440
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
1
799
943
087
230
373
515
657
799
940
*080
220
360
499
638
776
914
*051
188
325
461
596
732
866
*00 I
135
268
402
534
667
799
930.
*061
1
1
329
330
331
332
333
1
1
I
334
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
634 ' 647
763 776
892 905
53 020 033
148 161
275 288
403 415
529 542
656 668
.782 794
908 920
54 033 045
158 170
283 295
407 419
I
660
789
917
046
173
30 I
428
555
681
807
933
058
183
307
432
673
802
930
058
186
314
441
567
694
820
945
070
195
320
444
T - - -
686
815
943
071
199
326
453
580
706
832
958
083
208
332
456
- -"
699
827
956
084
212
339
466
593
719
845
970
095
22Q
345
469
192
1
Prop. Parts
813 I 828 842
958 972 986
101 116 130
244 259 273
387 401 416
530 544 558
671 686 700
813 827 841
954 968 982
*094 ' * 108 * 122
I 234 248 262
374 I 388 402
513 527 541
651 665 679
790 803 817
927 941 955
*065 *079 *092
202 215 229
338 352 365
474 488 501
610 623 637
745 759 772
880 893 907
*014 *028 *041
148 162 175
282 295 308
415 428 441
548 561 574
680 693 706
812 825 838
943 957 970
I
I
2
3
4
5
6
7
8
9
3.0
4.5
6.0
7.5
9.0
10.5
12.0
13.5
7r
=0.49715
9
349
362
I 479
492
1 t..'1
7
8
724 737 750
853 866 879
982 994 *007
110 122 135
237
250 263
364
377 390
491 504 517
618
631 643
744 757 769
870 882 895
995 *008 *020
120 133 145
245 258 270
370 382 394
494
506 518
9
*075 *088 * I 01
205 218 231
!
I
- -
I
1
2
3
4
5
6
7
8
9
363
364
365
366
367
368
369
370
14
1.4
2.8
4.2
5.6
7.0
8.4
9.8
1
2
3
4
5
6
7
8
1
2
3
4
5
6
323 I 336
- -711
840
969
097
224
352
479
605
732
857
983
108
233
357
481
log
1
1
15
1.5
,
I
I' 453 I 466
I
TABLE I
N. I L. 0
350 54 407
351
531
352
654
353
777
354
900
355 55 023
356
145
357
267
358
388
359
509
360
630
36'
751
362
871
1
371
11. 2
12.6
I
I
7.8
9 1
10.4
11.7
L.
0
2[3
0° 5'= 300" S. 4.68557 T. 4.68558
0 6 = 360
557
558
0
0
0
0
50
51
52
53
=3000
= 3060
= 3120
=3180
556
556
556
556
Prop. Parts
671891
561
561
561
561
0° 54'=3240"
0
0
0
0
0
55
56
57
58
59
= 3300
=3360
= 3420
= 3480
= 3540
S. 4.68556
556
556
555
555
555
I
I
2
432
555
679
802
925
047
169
291
413
534
654
775
895
*015
134
253
372
490
608
726
844
!
3
444
568
691
814
937
060
182
303
425
546
666
787
907
*027
146
265
384
502
620
738
855
I
4
066 078 089
183 194 206
299 310 322
415 426 438
530 542 553
646 657 669
761 772 784
875 887 898
990 *001 *013
104 115 127
218 229 240
331 343 354
444 456 467
557 i 569 i 580
659
670
681
387
388
771
883
782
894
794
906
390 59 106
391
218
392
329
393
439
394
550
395
660
396
770
397
879
398
988
399 60 097
400
206
N. I L. 0
118
229
340
450
561
671
780
890
999
108
217
129
240
351
461
572
682
791
901
*010
119
228
101
217
334
449
565
680
795
910
*024
138
252
365
478
i 591
69f 704
805 816
917 928
I
995 *006 *017 *028 *040
l2.J
2
140
251
362
472
583
693
802
912
*021
130
239
I
3
151
262
373
483
594
704
813
923
*032
141
249
561
561
561
562
562
0 58 = 3480
0 59 = 3540
1 0 =3600
555
555
555
6
I
7
I
8
I
9
481 494 506 518
605 617 630 642
728 741 753 765
851 864 876 888
974 986 998 *011
096 108 121 133
218 230 242 255
340 352 364 376
461 473 485 497
582 594 606 618
703 715 727 739
823 835 847 859
943 955 967 979
*062 *074 *086 *098
182 194 205 217
301 312 324 336
419 431 443 455
538 549 561 573
656 667 679 691
773 785 797 808
891 902 914 926
996 *008 *019 *031 *043
113 124 136 148 159
229 241 252 264 276
345 357 368 380 392
461 473 484 496 507
576 588 600 611 623
692 703 715 726 738
807 818 830 841 852
921 933 944 955 967
*035 *047 *058 *070 *081
149 161 172 184 195
263 274 286 297 309
377 388 399 410 422
490 501 512 524 535
602 I 614 I 1'>1<;
I I'>il'>
I 1'>47
715 726 7J7 749 760
827 838 850 861 872
939 950 961 973 984
*051 *062 *073 *084 *095
162 173 184 195 207
273 284 295 306 318
384 395 406 417 428
494 506 517 528 539
605 616 627 638 649
715 726 737 748 759
824 835 846 857 868
934 945 956 9/\6 977
*043 *054 *065 *076 *086
152 163 173 184 195
Prop. Parts
13
1
1.3
2
2.6
3
3.9
4
5.2
5
6.5
6
7.8
7 9. I
8 10.4
9 11.7
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
12
1.2
2.4
3.6
4.8
6.0
7.2
8.4
9.6
10.8
11
1. 1
2.2
3.3
4.4
5.5
6.6
7.7
8 I 8 8
~o
1
2
3
4
5
6
7
8
9
10
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
260 271 282 293 304
I~I
Prop. Parts
0° 5'= 300" S. 4.68557 T. 4.68558
0 6 = 360
557
558
0 7 = 420
557
558
T. 4.68561
I
5
456 469
580 593
704 716
827 839
949 962
072 084
194 206
315 328
437 449
558 570
678 691
799 811
919 931
*038 *050
158 170
277 289
396 407
514 526
632 644
750 761
867 879
937 949 961 972 984
386
389
1
N.
991
56 110
229
348
467
585
703
820
.
12
1.2
2.4
3.6
4.8
6.0
7.2
8.4
9.6
10.8
I
419
543
667
790
913
035
157
279
400
522
642
763
883
*003
122
241
360
478
597
714
832
372 57054
373
171
374
287
375
403
376
519
377
634
378
749
379
864
380
978
381 58 092
382
206
383
320
384
433
i 385
546 i
13
1.3
2.6
3.9
5.2
6.5
I
350.400
I
562
562
562
1
/
101'=3660"
1 2 3720
1 3 = 3780
=
1 4 = 3840
1 5 = 3900
1 6 = 3960
1 7=
4020
S. 4.68555 T. 4.68562
555
562
555
562
555
563
555
563
555
563
555
563
41
40
A
r--'
TABLE I
N. I L. 0
60 206
400
400-450
I
i
314
401
423
402
403
531
638
404
746
405
406
853
959
407
408 61 066
172
409
278
410
384
411
490
412
595
413
700
414
805
415
909
416
417 62 014
118
418
221
419
325
420
421
428
422
531
634
423
424
737
839
425
941
426
427 63 043
144
428
429
246
347
430
448
431
548
432
649
433
749
434
849
I 435
949
436
437 64 048
147
438
246
439
345
440
444
441
542
442
640
443
738
444
836
445
933
446
65 031
447
128
448
225
449
321
450
N.
L.
0
I~\
I
217
325
433
541
649
756
863
970
077
183
289
395
500
606
711
815
920
024
128
232
335
439
542
644
747
849
951
05Z
155
256
357
458
558
659
I 759
859
959
058
157
256
355
454
552
650
748
846
943
040
137
234
331
I
.
\
I
228
336
444
552
660
767
874
981
087
194
300
405
511
616
721
826
930
034
138
242
346
449
552
655
757
859
961
063
165
266
367
468
568
.i 669
769
869
969
068
167
266
365
464
562
660
758
856
953
050
147
244
341
2
I
249
260
I
271 1282
379 390
I
10
1 [1.0
2 2.0
3 3.0
4 4.0
5 5.0
6 6.0
7 7.0
8 8.0
9 9.0
1
2
3
4
5
6
7
8
9
I
I
\
4
6
5
7
8
I
9
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
I 483
! -is'}
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
11
1.1
2.2
3.3
4.4
5.5
6.6
7.7
8.8
9.9
1
2
3
4
5
6
7
8
9
I
3
TABLE I
N. I L.
~Prop.Parts
7
293 304
401 412
369
347
358
509 520
477
487 498
455 466
584 595 i 606 617 627
563 574
713 724 735
703
670 681 692
821 831 842
778 788 799 810
927 938 949
885 895 906 917
991 *002 *013 *023 *034 *045 *055
119 130 140 151 162
109
098
247 257 268
204 215 225 236
310 321 331 342 352 363 374
416 426 437 448 458 469 479
521 532 542 553 563 574 584
658 669 679 690
627 637 648
773 784 794
731 742 752 763
836 847 857 868 878 888 899
941 951 962 972 982 993 *003
107
076 086 097
045 055 066
190 201 211
149 159 170 180
252 263 273 284 294 304 315
356 366 377 387 397 408 418
459 469 480 490 500 511 521
603 613 624
562 \ 572 583 593
706 716 726
665 675 685 696
767 778 788 798 808 818 829
870 880 890 900 910 921 931
972 982 992 *002 *012 *022 *033
104 114 124 134
073 083 094
175 185 195 205 215 225 236
306 317 327 337
276 286 296
377 387 397 407 417 428 438
508 518 528 538
478 488 498
609 619 629 639
579 589 599
, 679 ,689
699. 709 . 719 729, 739
I 779 I 789 799 I 809 I 819 829 i 839
;
879 889 899 I 909 ! 919 ' 929 939
998' *008 1*018 (028 *038
979 988
108
118 128 137
078 088 098
227 237
177 187 197 207 217
306
316. 326 335
276 286 296
434
424
395 404 414
375 385
503 513: 523 532
483 493
473
631
601
611 621
572 582 591
699
709 719 729
689
670 680
807 816 826
787 797
777
768
895 904 914 924
885
875
865
992 *002 *011 *021
982
963 972
118
08
079
089
099
070
060
215
186
196 [I 205
176
167
157
312
302
273
283 292
263
254
389 398 408
369
379
350 360
239
0° 6'= 360" S. 4.68557 T. 4.68558
558
557
0 7 = 420
558
557
0 8 = 480
563
555
1 6 = 3960
563
555
1 7 = 4020
563
555
1 8 =4080
- 42
\
6
I
9
0.9
1.8
2.7
3.6
4.5
5.4
6.3
7.2
8.1
Prop. Parts
N.
1° 9'=4140" S. 4.68555
563
554 T. 4.68563
1
1
1
1
1
1
10 =4200
11 =4260
12 =4320
13 = 4380
14 =4440
15 =4500
554
554
554
554
554
564
564
564
564
564
0°
0
0
..
450-500
65 321
I
418
427
514 I 523
610 619
706
715
801
811
896
906
992 *001
096
66 087
181
191
276 285
370 380
464 474
558 567
652 661
745
755
839 848
932
941
034
67 025
117 127
210
219
302
311
394 403
495
486
578 587
669 679
761
770
852 861
943
952
043
68 034
124
133
224
215
305 . 314
395 I 404
.F"
OJ j '}94
.
I
574
583
664
753
842
931
69 020
108
197
285
373
461
548
636
723
810
897
673
762
851
940
028
117
205
294
381
469
557
644
732
819
906
L.
0
I
7'= 420"
8 = 480
9 = 540
1 15 =4500
1 16 =4560
1 17 =4620
I
Prop. Parts
6
7
8
9
369
466
562
658
753
849
944
*039
134
229
323
417
511
605
699
792
885
978
071
164
256
348
440
532
624
715
806
897
988
079
169
260
350
440'
.529
619
708
797
886
975
064
152
241
329
417
504
592
679
767
854
940
379
475
571
667
763
858
954
*049
143
238
332
427
521
614
708
801
894
987
080
173
265
357
449
541
633
724
815
906
997
088
178
269
359
I 449
538
!
628
717
806
895
984
073
161
249
338
425
513
601
688
775
862
949
389
485
581
677
772
868
963
*058
153
247
342
436
530
624
717
811
904
997
089
182
274
367
459
550
642
733
825
916
*006
097
187
278
368
.
I 458
547
398
495
591
686
782
877
973
*068
162
257
351
445
539
633
727
820
913
*006
099
191
284
376
468
560
651
742
834
925
*015
106
196
287
377
. 467
I
5.56
I
646
735
824
913
*002
090
179
267
355
443
531
618
705
793
880
966
408
504
600
696
792
887
982
*077
172
266
361
455
549
642
736
829
922
*015
108
201
293
385
477
569
660
752
843
934
*024
115
205
296
386
. 476
I 565
I 655
744
833
922
*011
099
188
276
364
452
539
627
714
801
888
975
I
I
I
I
I
I
2
331
I
350 360
341
447 456
437
543 552
533
629 639 648
725
734 744
820 830 839
925 935
916
*011 *020 *030
106 115 124
200 210 219
304 314
295
398 408
389
492 502
483
577 586 596
671 680 689
764 773 783
857 867 876
950 960 969
043 052 062
136
145 154
228
237 247
321
330 339
413
422 431
504 514 523
596
605 614
688
697 706
788 797
779
870
879 888
970 979
961
052 061 070
142 151 160
233
242 251
323 332 341
. 413
I
I 422 :! 431
520
I) "02 j 5'1
I
I
I 592 I 601
610
681 690 699
771 78(J 789
860 869 878
949 958 966
046 055
037
126 135 144
214
223 232
302
311 320
390 399 408
478 487 496
583
566 574
671
653 662
740
749 758
827
836 845
932
914 923
I
0
4
3
5
.
2
I
3
I
4
S. 4.68557 T. 4.68558
557
558
558
557
554
554
554
564
565
565
5
1°
1
1
1
1
6
i 637
726
815
904
993
082
170
258
346
434
522
609
697
784
871
958
7
18'=4680"
19 =4740
20 = 4800
21 =4860
22 = 4920
I
23 = 4980
1 24 =5040
8
9
10
1 1.0
2 2.0
3 3.0
4 4.0
5 5.0
6 6.0
7 7.0
8[8.0
9 9.0
I
1
2
3
4
5
6
7
8
9
9
0.9
1.8
2.7
3.6
4.5
5.4
6.3
7.2
8.1
1
2
3
4
5
6
7
8
9
8
0.8
1.6
2.4
3.2
4.0
4.8
5.6
6.4
7.2
Prop. Parts
S. 4.68 554 T. 4.68565
565
554
565
554
566
553
566
553
566
553
566
553
43
~
TABLE I
N.
I L.
500-550
I
I
7
8
I
9
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
69 897
984
70 070
157
243
329
415
501
586
672
757
842
927
71 012
096
\81
265
349
433
517
600
684
767
850
933
72 016
099
181
263
346
428
509
591
536
537
538
539
640
541
542
543
544
545
546
547
548
549
550
852
86U ~o~ 876 884 892 900 908
916
925 933
941 949
957 965
973 981 989
997 *006 *014 *022 *030 *038 *046 *054 *062 *070
73 078 086 094
102 111 119
127 135 143 151
159 167
175 183 191 199 207 215 223 231
239 247 255
263 272
280 288 296
304 312
320 328 336
344 352 360
368 376 384 392
400 408 416
424 432 440 448 456
464 472
480
488 496
504 512 520 528
536 544 552
560 568
576 584 592
600 608
616 624 632
640
648 656
664 672 679. I 687 695
703 711
719 727
735 743
751 759: 767
775 783 791
799 807
815 823 830 838 i 846 854
862 870
878
886 894 902
910 918 i 926 933
941 949
957 965
973 981 989 997 *005 *013 *020 *028
74 036 ,044
052
060 068 076
084
092
099
107
N.
I
L.
00 8'=
0
I
I
480"
0 9 = 540
0 10 = 600
1 23 =4980
1 24 =5040
1 25 =5100
44
906
914 923
932
992 *001 *010 *018
079
088 096
105
165
174 183 191
252 260
269 278
338
346 355 364
424 432 441 449
509 518 526 535
595
603 612 621
680
689 697 706
766 774 783
79\
851
859 868 876
935 944
952 961
020
029 037 046
105
113 122 130
189 198 206 214
273 282 290 299
357 366
374 383
441 450 458 466
525 533 542 550
609 617
625 634
692
700 709 717
775
784 792 800
858
867 875 883
941 950
958 966
024 032
041 049
107
115 123 132
189
198 206 214
272
280 288 296
354
362 370 378
436
444 452 I 460
518
526 534
542
599
607 616 I ~?~
I
I
2
I
S. 4.68557
557
557
553
553
553
I
3
I
I
958 I 966
975'
!
*027 *036 *044 *053 *062
114 122
131
140 148
200 209 217
226
234
286 295 303
312 321
372 381 389
398 406
458 467
475 484
492
544 552
561 569
578
629
638 646 655
663
714 723 731
740 749
800 808
817 825 834
885 893 902
910 919
969 978 986
995 *003
054
063 071
079 088
139 147
155
164 172
223 231 240
248 257
307 315 324
332 341
391 399 408 416
425
475 483
492
500 508
559 567 575 584 592
659 667
642 650
675
742 750 759
725 734
809 817 825 834 842
892 900 908 917 925
975
983 991 999 *008
074 082 090
057 066
140 148 156 165 173
222 230 239 247 255
304 313 321 329 337
387 395 403
411 419
469 477 485
493 501
550 558 567
575
583
~3?I 640 648 656 665
Prop. Parts
9
1
2
3
4
5
6
7
8
9
I
0.9
1.8
2.7
3.6
4.5
5.4
6.3
7.2
8.1
8
1
2
3
4
5
6
7
8
9
I 0.8
1.6
2.4
3.2
4.0
4.8
5.6
6.4
7.2
I
1
2
3
4
5
6
7
8
9
1
0.7
1.4
2.1
2.8
3.5
4.2
4.9
5.6
6.3
N.
I
5
T. 4.68558
558
558
566
566
566
1
1
I
6
I
7
I
1026'=5160"
\ 27 = 5220
1 28 = 5280
1 29 = 5340
1 30 = 5400
1 31 = 5460
1 32 = 5520
8
I
9
I L.
I
Parts
S. 4.68 553 T. 4.68567
553
567
553
567
553
567
553
567
552
568
552
568
I
I
I
U
I
2
060
139
218
296
374
453
531
609
687
764
842
920
997
074
151
228
305
381
458
534
610
686
762
838
914
989
065
140
215
290
365
44Q
582
492
500
507
515
586
587
588
589
590
591
592
593
594
595
596
597
598
599
790
864
938
77 012
085
159
232
305
379
452
525
597
670
743
797
871
945
019
093
166
240
313
386
459
532
605
677
750
805
879
953
026
100
173
247
320
393
466
539
612
685
757
812
885
960
034
107
181
254
327
401
474
546
619
692
764
N.
I
3
052
131
210
288
367
445
523
601
679
757
834
912
989
066
143
220
297
374
450
526
603
679
755
831
906
982
057
133
208
283
358
433
4
516171819
076 084 092 099 107
155 162 170 178 186
233 241 249 257 265
312 320 327 335 343
390 398 406 414 421
468 476 484 492 500
547 554 562 570 578
624 632 640 648 656
702 710 718 726 733
780 788 796 803 811
858 865 873 881 889
935 943 950 958 966
*012 *020 *028 *035 *043
089 097
105 113
120
166 174 182
189
197
243 251 259 266
274
320 328 335
343 351
397 404 412
420 427
473 481 488
496
504
549 557 565
572
580
626 633 641
648 656
702 709 717
724
732
778 785 793 800
808
853 861 868
876 884
929 937 944
952 959
*005 *012 *020 *027 *035
080 087 095
103 110
155 163
170 178 185
230 238 245
253 260
305 313 320
328 335
380 388 395 403
410
448 455 462
470 477 485
522 530 537 545 552 559
068
147
225
304
382
461
539
617
695
772
850
927
*005
082
159
236
312
389
465
542
618
694
770
846
921
997
072
148
223
298
373
I
I
L.
0
I
I
I
2
819
893
967
041
115
188
262
335
408
481
554
627
699
772
827
901
975
048
122
195
269
342
415
488
561
634
706
779
~
31
32
33
34
=5460
=5520
=5580
=5640
552
552
552
552
834
908
982
056
129
203
276
349
422
495
568
641
714
786
842
916
989
063
137
210
283
357
430
503
576
648
721
793
7
00 9'= 540" S. 4.68557 T. 4.68558
0 10 = 600
557
558
1
1
1
1
Prop. Parts
1
2
3
4
5
6
7
8
9
8
0.8
1.6
2.4
3.2
4.0
4.8
5.6
6.4
7.2
5
6
7
8
9
3.5
4.2
4.9
5.6
6.3
I
849
923
997
070
144
217
291
364
437
510
583
656
728
801
815 822 830 837 844 851 859 866 873
600
Prop.
0
550 74 036 044
551
115 123
552
194 202
553
273 280
554
351 359
555
429 437
556
507 515
557
586 593
558
663 671
559
741 749
560
819 827
561
896 904
562
974 981
563 75 051 059
564
128 136
565
205 213
566
282 289
567
358 366
568
435 442
569
511 519
570
587 595
571
664 671
572
740 747
573
815 823
574
891 899
575
967 974
576 76 042 050
577
118 125
578
193 200
579
268 275
580
343 350
581
418 125
1
14
550-600
TABLE I
I 6
940, 949
~4 I 5
0
568
568
568
568
1
I
1035'=5700"
1 36 = 5760
1 37 = 5820
I 38 = 5880
J 39 =5940
1 40 =6000
8
856
930
*004
078
151
225
298
371
444
517
590
663
735
808
880
L2..J
Prop. Parts
S. 4.68552 T. 4.68569
552
569
552
569
552
569
551
569
551
570
45
l
TABLE I
N. I L.
600
60 I
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
I
0
I
I
2
I
77 815 822 830
887 895 902
960 967 974
78 032 039 046
104 111 118
176 183 190
247 254 262
319 326 333
390 398 405
462 469 476
533 540 547
604 611 618
675 682 689
746 753 760
817824831
888 895 902
958 965 972
79 029 036 043
099 106 113
169 176 183
239 246 253
309 316 323
379 386 393
449 456 463
518 525 532
588 595 602
657 664 671
727 734 741
796 803 810
865 872 879
934 941 948
80 003 010 017
024
092
072
085
634
1f)9 i 1111 I. 223 . 229
635
277
636
637
638
639
640
641
642
643
644
645
646
647
346
414
482
550
618
686
754
821
889
956
81 023
090
648
649
650
N.
4
7
I
I
Prop. Parts
9
030
099
037 044 051 058 065
10§ 113 120 127 134
236
243.
, ,uo
'U I I
I J"
1/
I
I
298
353
421
489
557
625
693
760
828
895
963
030
366 373
434 441
502 509
570 577
638 645
706 713
774 781
841 848
909 916
I
976 983
043 050
]
2
3
4
5
6
7
8
9
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
8
0.8
1. 6
2.4
3.2
4.0
4.8
5.6
6.4
7.2
1
7
0.7
1. 4
2.1
2.8
3.5
1
2
3
4
5
6
7
8
9
N.
1
4.2
4.9
5.6
6.3
100
I ~u
"0":;
1 250
I'. 257
1. 264
1. 271
..:;u..:;
312 I 318 i 325
305
284! 291
359
428
496
564
632
699
767
835
902
969
037
/ .J
380
448
516
584
652
720
787
855
922
990
057
i
332 i 339
387 393 400 407
455 462 468 475
523 530 536 543
591 598 604 611
659 665 672 679
726 733 740 747
794 801 808 814
862 868 875 882
929 936 943 949
996 *003 *010 *017
064 070 077 084
1
2
3
4
5
6
7
8
9
, rQi
i
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
6
0.6
1.2
1.8
2.4
3.0
3.6
4.2
4.8
5.4
097 104 111 117 124 131 137 144 151
158164171178184191198204211218
224 231 238 245 251 258 265 271 278 285
291 298 305 311 318 325 331 338 345 351
0
I
I
I
2
I
3
I
4
I
0° 10'= 600" S. 4.68557 T. 4.68 558
557
558
0 11 = 660
551
570
1 40 = 6000
551
570
1 41 =6060
570
551
1 42 =6120
551
570
1 43 =6180
I
5
1
6
!
7
I
JO 44'=6240"
1 45 =6300
1 46 =6360
1 47 = 6420
I 48 = 6480
1 49 = 6540
8
I
9
I
Prop.
I L.
I
56
13m
I
81 291 298 305 311 318 325 331
358 365 371 378 385 391 398
425 431 438 445 451 458 465
491 498 505 511 518 525 531
558 564 571 578 584 591 598
624 631 637 644 651 657 664
690 697 704 710 717 723 730
757 763 770 776 783 790 796
823 829 836 842 849 856 862
889 895 902 908 915 921 928
954 961 968 974 981 987 994
82 020 027 033 040 046 053 060
086 092 099 105 112 119 125
151 158 164 171 178 184 191
217 223 230 236 243 249 256
282 289 295 302 308 315 321
347 354 360 367 373 380 387
413 419 426 432 439 445 452
478 484 491 497 504 510 517
543 549 556 562 569 575 582
607 614 620 627 633 640 646
672 679 685 692 698 705 711
737 743 750 756 763 769 776
802 808 814 821 827 834 840
866 872 879 885 892 898 905
930 937 943 950 956 963 969
995 *001 *008 *014 *020 *027 *033
83 059 065 072 078 085 091 097
123 129 136 142 149 155 161
187 193 200 206 213 219 225
251 257 264 270 276 283 289
315 321 327 334 340 347 353
378 385 391 398 404 410 417
4))
41)U
461
46/
4/4
'I'tT 1'f'fO "I
~"h ~i1
~iQ 1 ~1~ ,,1
~17. 1 <;H
i
i
i
i
569 575 582 588 594 601 1 607
632 639 645 651 658 664 670
696 702 708 715 721 727 734
759 765 771 778 784 790 797
822 828 835 841 847 853 860
885 891 897 904 910 916 923
948 954 960 967 973 979 985
84 011 017 023 029 036 042 048
073 080 086 092 098 105 111
136142148155161167173180186192
198 205 211 217 223 230 236
261 267 273 280 286 292 298
323 330 336 342 348 354 361
386 392 398 404, 410 417 423
448 454 460 466 473 479 485
'
0
I
I
2
I
700
510 I,516
N. IL. 0
I
I L.
650-700
TABLE I
8
837 84'., 851 859 866 873 880
909 91,,' 924 931 938 945 952
981 988 996 *003 *010 *017 *025
053 061 068 075 082 089 097
125 132 140 147 154 161 168
197 204 211 219 2L6 233 240
269 276 283 290 297 305 312
340 347 355 362 369 376 383
412 419 426 433 440 447 455
483 490 497 504 512 519 526
554 561 569 576 583 590 597
625 633 640 647 654 661 668
696 704 711 718 725 732 739
767 774 781 789 796 803 810
838845852859866873880
909 916 923 930 937 944 951
979 986 993 *000 *007 *014 *021
050 057 064 071 078 085 092
120 127 134 141 148 155 162
190 197 204 211 218 225 232
260 267 274 281 288 295 302
330 337 344 351 358 365 372
400 407 414 421 428 435 442
470 477 484 491 498 505 511
539 546 553 560 567 574 581
609 616 623 630 637 644 650
678 685 692 699 706 713 720
748 754 761 768 775 782 789
817 824 831 837 844 851 858
886 893 900 906 913 920 927
955 962 969 975 982 989 996
632
I
079
3
600-650
I 5 I 6 I
'
522
528
I
535
541
547
I
7
242
305
367
429
491
553
~
Parts
0° 10'=
S. 4.68 551 T. 4.68571
551
571
551
571
550
572
550
572
550
572
600"
0 11 = 660
0 12 = 720
1 48 =6480
1 49 =6540
I 50 =6600
46
I
8
248
311
373
435
497
559
8
S. 4.685.17 T. 4.68558
557
557
558
558
550
550
550
572
572
572
1
1
I
Prop. Parts
9
338 345 351
405 411 418
471 478 485
538 544 551
604 611 617
671 677 684
737 743 750
803 809 816
869 875 882
935 941 948
*000 *007 *014
066 073 079
132 138 145
197 204 210
263 269 276
328 334 341
393 400 406
458 465 471
523 530 536
588 595 601
653 659 666
718 724 730
782 789 795
847 853 860
911 918 924
975 982 988
*040 *046 *052
104 110 117
168 174 181
232 238 245
296 302 308
359 366 372
423 429 436
41)
'I 41)/ 1~
. <;<;". ,,;r. 1
.
I
i
! "
613 620 626
677 683 689
740 746 753
803 809 816
866 872 879
929 935 942
992 998 *004
055 061 067
117 123 130
1051'=6660"
1 52 = 6720
1 53 = 6780
I 54 = 6840
1 55 = 6900
I 56 = 6960
1 57 = 7020
7
0.7
1.4
2.1
2.8
3.5
4.2
4.9
5.6
6.3
1
2
3
4
5
6
7
8
9
6
0.6
H-t-:-2.
~1
4 I 2.4
5
6
7
8
9
3.0
I
1
3.6
4.2
4.8
5.4
255
317
379
442
504
566
19
I
Prop. Parts
S. 4.68550 T. 4.68573
550
573
550
573
550
573
549
574
549
574
549
574
47
.&
l
TABLE I
I L.
N.
700
I
0
I
I
2
I
3
I
4
700.750
I 5 I 6 I
7
I
8
I
Prop. Parts
9
84 510
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
N.
516 522 528 535 541 '547
553
559 566
572 578 584 I 590
597 ,.603
609 615 62\
628
634
640
646 652 658 665 671 677 683 689
696
702
708 714 720 726 733 739
745 751
757
763
770 776 782 788 794 800 807 813
819
825 831 837 844 850
856 862 868 874
880
887 893 899 905 911 917
924 930 936
942
948 954 960 967 973 979 985 991 997
85 003
009 016 022 028 034
040 046 052 058
065 071 077 083 089 095
101
107 114 120
126
132 138 144 150
156 163
169 175 181
187
193
199 205 211 217 224 230 236 242
248 254 260 266 272
278 285 291 297 303
309 315
321 327 333 339 345 352 358 364
370
376 382 388 394 400
406 412 418 425
431 437
443 449 455 461
467 473 479 485
491 497
503 509 516 522
528 534 540 546
552
558 564 570 576 582
588 594 600 606
612
618 625 631 637 643 649 655
661 667
673 679 685 691 697
703 709 715 721 727
733 739 745 751 757
763 769 775 781 788
794 800 806 812 818 824 830
836 842 848
854
860 866 872 878 884
890 896 902 908
914
920 926 932 938 944
950 956 962 968
974 980 986 992 998 *004 *.0 10 *016 *022 *028
86 034 040 046 052 058 064 070 076 082 088
094
100
106 112 118 124
130 136 141 147
153 159 165 171 177 183
189 195 201 207
213 219
225 231 237 243
249 255
261 267
273 279 285 291 297 303 308 314 320 326
332
338 344 350 356 362 368 374 380 386
392
398 404 410 415 421
427 433 439 445
451 457 463 469 475 481
487 493 499 504
510 516 522 528 534 540 546' 552
558 564
570
576 581 587 593 599 605 611 617 623
I
I
I
I
I
I
I
629 635 641 646 652 658! 664 670 676 682
688694700705711717723729735741
747 753 759 764 770 776 782 788
794 800
806
812 817 823 829 835 841 847 853 859
864
870 876 882 888 894 900 906
911 917
923
929 935 941 947
953 958 964 970 976
982 988
994 999 *005 *011 *017 ¥023 *029 *035
87 040 046 052 058 064 070
075 081 087 093
099
105 111 116 122 128 134 140
146 151
157
163 169 175 181 186192
198204210
216
221 227 233 239 245
251 256 262 268
274
280 286 291 297 303 309 315
320 326
332 338 344 349 355 361 36Z 373
379 384
390 396 402 408 413 419
425 431 437 442
448 454
460 466 471
477 483 489
495 500
506
512 518 523 529 535
541, 547
552 558
1
.
7
0.7
1.4
2.1
2.8
3.5
4.2
4.9
5.6
6.3
1
2
3
4
5
6
7
8
9
6
0.6
1.2
I. 8
2.4
3.0
3.6
4.2
4.8
5.4
1
1
2
3
4
5
6
7
8
9
5
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
1
L.
0
I
2
3
I
4
5
0° 11'= 660" S. 4.68557 T. 4.68558
557
558
0 12 = 720
557
558
0 13 = 780
574
549
1 56 =6960
574
549
1 57 ~7020
549
575
I 58 = 7080
-
48
1
2
3
4
5
6
7
8
9
6
7
1° 59'=7140"
2 0 =7200
2 1 =7260
2 2 =7320
2 3 = 7380
2 4 = 7440
2 5 = 7500
8
9
T
Prop. Parts
S. 4.68549 T. 4.6857;
549
575
549
575
548
576
548
548
548
576
576
577
TABLE I
N. L.
750.800
0
I
2
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
87 506
564
622
679
737
795
852
910
967
88 024
081
138
195
252
309
366
423
480
536
593
649
705
762
818
874
930
986
89 042
098
154
209
265
321
376
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
487 I 492 I 498
542 548
553
597
603
609
653 658
664
708 713
719
763 768
774
818 823
829
873 878
883
927 933
938
982 988 993
90 037 042
048
091 097
102
146 151 157
200 206 211
255 260 266
309 314
320
N.
L.
0
3
4
6
5
8
7
I
9
523 529 535 541 547
552 558
5121518
570 576
581 587 593 599
604
610 616
628
633 639 645 651 656 662 668
674
708 714 720
685
691 697 703
726 731
743 749 754 760 766 772 777 783
789
800 806
812 818 823 829 835
841 846
858 864 869 875 881 887 892 898 904
915 921 927 933 938 944
950 955 961
973 978 984 990 996 *001 *007 *013 *018
030 036 041 047 053 058 064
070 076
087 093 098
104 110 116 121 127
133
144 150
156 161 ]67
173 178 184
190
201 207
213 218 224 230 235
241 247
258 264
270 275 281 287 292
298
304
315 321
326 332 338 343 349 355
360
372 377
383 389 395 400 406 412
417
429 434 440 446 451 457 463 468
474
485 491 497 502 508 513 519 525
530
542 547 553
559 564 570 576
581 587
598
604 610 615 621 627 632
638 643
655
660 666 672 677 683 689 694
700
711 717 722 728 734 739 745 750 756
767 773
779 784 790 795 801 807 812
824
829 835 840 846 852 857
863 868
880
885 891 897 902 908 913
919 925
936
941 947 953 958 964 969 975
981
992
997 *003 *009 *014 *020 *025 *031 *037
059 064 070 076 081 087 092
048 053
104
109 115 120 126 131 137 143 148
170 176 182 187 193 198 204
159 165
215
221 226 232 237 243 248 254 260
282
287 293 298 304 310 315
271 1276
337 343 348 354 360
326 332
365 371
393 398 404 409 415, 421 426
382 387
Prop.
Parts
1
2
3
4
5
6
7
8
9
6
0.6
1.2
1.8
2.4
3.0
3.6
4.2
4.8
5.4
I
2
515 i 520 I 526 ! 531 I 537
559 564 570 575 581 586
592
614 620 625 631 636
642 647
669
675 680 686 691
697 702
724 730 735 741 746
752 757
779 785 790 796 801 807 812
834 840 845 851
856 862 867
889 894 900 905 911
916 922
944
949 955 960 966 971 977
998 *004 *009 *015 *020 *026 *031
053 059 064 069 075 080 086
108 113 119 124 129
135 140
162 168 173 179 184
189 195
217 222 227 233 238
244 249
271 276 282 287 293
298
304
325 331 336 342 347
352 358
! 504 I 509
3
4
0" 12'= 720" S. 4.68557 T. 4.68 558
0 13 = 780
557
558
0 14 = 840
557
558
2 5 =7500
548
577
2 6 =7560
548
577
2 7 =7620
548
577
5
I
6
I
7
2° 8'=7680"
2 9 = 7740
2 10 =7800
2 11 =7860
1
2 12 =7920
2 13 =7980
2 14 = 8040
I
8
I
9
I
Prop. Parts
S. 4.68 547 T. 4.68578
547
578
547
547
547
547
546
578
579
579
579
579
49
l
TABLE I
N.
I
800-850
L.
0
I
I
I
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
90 309
363
417
472
526
580
634
687
741
795
849
902
956
91 009
062
116
169
222
275
328
381
434
487
540
593
645
698
751
803
855
908
960
314
369
423
477
531
585
639
693
747
800
854
907
961
014
068
121
174
228
281
334
387
440
492
545
598
651
703
756
808
861
913
965
832
833
92 012
.0651
018
010
--'--1334
--
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
N.
nTlm
169
221
273
324
376
428
480
531
583
634
686
737
788
840
891
942
I L.
0
t
2
I
I
3
I
I
I
TABLE I
I
5
6
320
374
428
482
536
590
644
698
752
806
859
913
966
020
073
126
180
233
286
339
392
445
498
551
603
656
709
761
814
866
918
325
380
434
488
542
596
650
703
757
811
865
918
972
025
078
132
185
238
291
344
397
450
503
556
609
661
714
766
819
871
924
331
385
439
493
547
601
655
709
763
816
870
924
977
030
084
137
190
243
297
350
403
455
508
561
614
666
719
772
824
876
929
336
390
445
499
63
607
660
714
768
822
875
929
982
036
089
142
196
249
302
355
408
461
514
566
619
672
724
777
829
882
934
971
023
075
976
028
oan
981
033
illl5.
986 I 991
038. 044
r17
I).l
i 174 I 179
226
278
330
381
433
485
536
588
639
691
742
793
845
896
947
4
184 I
231 236
283 288
335 340
387 392
438 443
490 495
542 547
593 598
645 650
696 701
747 752
799 804
850 855
901 906
952 957
I
342
396
450
504
558
612
666
720
773
827
881
934
988
041
094
148
201
254
307
360
413
466
519
572
624
677
730
782
834
887
939
7
I
347
401
455
509
563
617
671
725
779
832
886
940
993
046
100
153
206
259
312
365
418
471
524
577
630
682
735
787
840
892
944
8
1
2
3
4
5
6
7
8
9
091 ,096
1
1
151
In1T48TIT
997 *002 *007
049 054 059
101 .106 H!
1
~-1511--1H
189
241
293
345
397
449
500
552
603
655
706
758
809
860
911
962
195
247
298
350
402
454
505
557
609
660
711
763
814
865
916
967.
205!
257
309
361
412
464
516
567
619
670
722
773
824
875
927
978
200
252
304
355
407
459
511
562
614
665
716
768
819
870
921
973
N.
Prop. Parts
9
358
412
466
520
574
628
682
736
789
843
897
950
*004
057
110
164
217
270
323
376
429
482
535
587
640
693
745
798
850
903
955
'
352
407
461
515
569
623
677
730
784
838
891
945
998
052
105
158
212
265
318
371
424
477
529
582
635
687
740
793
845
897
950
I
-,
210 ! 215
262 267
314 319
366 371
418 423
469 474
521 526
572 578
624 629
675 681
727 732
778 783
829 834
881 886
932 937
983 988
I
.
2.0
2.5
3.0
3.5
4.0
4.5
Prop.
Parts
L.
0
2
I
3
I
4
I
0013'= 780" S. 4.68557 T. 4.68 558
557
558
0 14 = 840
557
558
0 15 = 900
547
579
2 13 = 7980
546
579
2 14 =8040
546
580
2 15 =8100
5
1
/
I
6
I
7
I
2016'=8160"
2 17 =8220
2 18 =8280
2 19 = 8340
2 20 = 8400
2 21 =8460
2 22 = 8520
8
I
9
I
N.
S. 4.68 546 T. 4.68580
546
580
546
581
546
581
545
582
545
582
545
582
2
3
I
4
I
I
L.
0° 14'=
015=
2
2
2
2
I
I
I
I
I
850-900
6
I 5
92 942 947 952 957 962
993 998 *003 *008 *013
93 044 049 054 059 064
095 100 105 110 115
146 151 156 161 166
197 202 207 212 217
247 252 258 263 268
298 303 308 313 318
349 354 359 364 369
399 404 409 414 420
450 455 460 465 470
500 505 510 515 520
551 556 561 566 571
601 606 611 616 621
651 656 661 666 671
702 707 712 717 722
752 757 762 767 772
802 807 812 817 822
852 857 862 867 872
902 907 912 917 922
952 957 962 967 972
94 002 007 012 017 i 022
052 057 062 067 072
101 106 111 116 121
151 156 161 166 171
201 206 211 216 221
250 255 260 265 270
300 305 310 315 320
349 354 359 364 369
399 404 409 4141419
448 453 458 463 468
498 503 507 512 517
547 552 557 562 567
I
596 6!llfill6T.nll...
612
04J -os-o- 655 660'.(jQJ.
j
694 699 . 704 ! 709 714
758 763
743 ' 748
753
792 797 802 807. 812
841 846 851 852: 861
890 895 900 905 910
939 944 949 954, 959
988 993 998 *002 . *007
95 036 041 046 051 056
085 090 095 100 105
134 139 143 148 153
182 187 192 197 202
231 236 240 245 250
279 284 289 294 299
328 332 337 342 347
376 381 386 390 395
424 429 434 439 444
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
R84
I 885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
6
0.6
1.2
1.8
2.4
3.0
3.6
4.2
4.8
5.4
4
5
6
7
8
9
I
21
22
23
24
0
I
840"
900
=8460
=8520
=8580
=8640
I
I
2
I
3
I
4
U/U
719
j
I
UI J
UOU
724 !I 729
768 773 778
817 822 827
866 871 876
915 919 924
963 968 973
*012 *017 *022
061 066 071
109 114 119
158 163 168
207 211 216
255 260 265
303 308 313
352 357 361
400 405 410
448 453 458
!
S. 4.68557 T. 4.68558
558
557
545
545
545
545
7
8
9
967 973 978 983 988
*018 *024 *029 *034 *039
069 075 080 085 090
120 125 131 136 141
171 176 181 186 192
222 227 232 237 242
273 278 283 288 293
323 328 334 339 344
374 379 384 389 394
425 430 435 440 445
475 480 485 490 495
526 531 536 541 546
576 581 586 591 596
626 631 636 641 646
676 682 687 692 697
727 732 737 742 747
777 782 787 792 797
827 832 837 842 847
877 882 887 892 897
927 932 937 942 947
977 982 987 992 997
027 032 037 042 047
077 082 086 091 096
126 131 136 141 146
176 181 186 191 196
226 231 236 240 245
275 280 285 290 295
325 330 335 340 345
374 379 384 389 394
424 429 433 438 443
473 478 483 488 493
522 527 532 537 542
571 576 581 586 591
621 626 630 635 640
582
582
583
583
5
I
2°
2
2
2
2
2
6
25'=8700"
26 =8760
27 =8820
28 =8880
29 =8940
30 =9000
UOJ
j 00""
734 738
783 787
832 836
880 885
929 934
978 983
*027 *032
075 080
124 129
173 177
221 226
270 274
318 323
366 371
415 419
463 468
I
Prop. Parts
6
0.6
1.2
1.8
2.4
3.0
3.6
4.2
4.8
5.4
1
2
3
4
5
6
7
8
9
5
1 0.5
2 1.0
.
3 1.5
4 2.0
5 2.5
6 3.0
7 3.5
8 4.0
9 4.5
I
------1-
1
2
3
4
5
6
7
8
9
4
0.4
0.8
1.2
1.6
2.0
2.4
2.8
3.2
3.6
~Prop.Parts
S. 4.68 545 T. 4.68 583
584
544
584
544
584
544
585
544
585
544
51
60
..
900.950
TABLE I
N.
L.
2
I
3
4
5
:
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
7
8
:
;
I
I
L.
0
I
:3
3
I
4
5
0015'= 900" S. 4.68557 T. 4.68 558120
557
558 2
0 16 = 960
2
544
585 2
2 30 =9000
585 2
544
2 31 =9060
586 2
543
2 32 =9120
586
543
2 33 ~9180
950.1000
TABLE I
6
458
463
439
444
448
453
95 424 4291434
506 511
492
497 50\
472
477 482 487
554
559
545
550
530 535 540
521 525
602
607
593 598
574
578 583 588
569
650 655
622 626 631 636
641 646
617
689
694 698
703
670 674 679 684
665
746 751
722 727 732 7 f 742
713 718
794 799
789
770 775 780 785
761 766
842
847
832 837
809 813 818 823 828
890 895
866 871 875 880 885
856 861
942
933 938
904
909
914 918 923 928
976
990
980 985
952 957
96\
966 971
999 *004 *009 *014 *019 *023 *028 *033 *038
076 080
085
071
96 047 052
057 061 066
123
128
133
095 099
104 109 114 118
171 175
180
142 147
152 156 161 166
223
227
218
190
194 199 204 209 213
265 270 275
237
242 246 251 256 261
308
313 317
322
284
294 298 303
289
360 365
369
332 336
341 346 350 355
412
417
407
388 393 398 402
379 384
459 464
454
435 440 445 450
426 431
506
501
511
473
478
483 487 492 497
548
553
558
520 525
530 534 539 544
600 605
595
577 581 586
59\
567 572
647 652
642
614
619 624 628 633 638
694
689
699
661
666
670 675 680 685
741
745
717 722
727 731 736
708 713
783 788
792
764 769 774 778
755 759
834
830
839
802 806 811 816 820 825
872 876
881
886
848 853
858 862 867
923
928
932
904 909 914 918
895 900
974 979
942 946
951 956 960 965 970
'*
*
*
*
*
97 033
039 044 049 ; 053 058 I 063 ; 067 072;
I
109
114
118
081 '086
090
095
100 104
160
155
165
128 132 137 142, 146 151
206
211
174
179 183 188 192 197 202
253 257
243 248
220 225 230 234 239
299
304
290 294
267 271 276 280 285
340
345
350
313 317 322 327 331 336
387
391
396
359 364 368 373 377 382
442
433
437
405 410 414 419 424 428
483
488
474 479
451 456 460 465 470
525
529 534
502 506 511 516 520
497
580
571
575
552 557 562 566
543 548
617 621
626
589 594 598 603 607 612
663 667
672
635 640 644 649 653 658
708
713
717
690 695 699 704
681 685
754
759 763
727 731 736 740 745 749
800 804! 809
772 777 782 786 791 795
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
N.
0
6
7
34'=9240"
35 = 9300
36 =9360
37 =9420
38 =9480
39 =9540
8
Prop. Parts
9
468
516
564
612
660
708
756
804
852
899
947
995
*042
090
137
185
232
280
327
374
421
468
515
562
609
656
703
750
797
844
890
937
984
*
077
I
1
2
3
4
5
6
7
8
9
5
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
N.
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
i ;04
123
985
169
216
262
308
354
400
447
493
539
585
630
676
722
768
813
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
9
I
Prop. Parts
S. 4.68 543 T. 4.68587
543
587
587
543
588
542
588
542
588
542
L.
0
1
97 772
818
864
909
955
98 000
046
091
137
182
227
272
318
363
408
453
498
543
588
632
677
722
767
811
856
900
945
989
99 034
078
123
167
211
i
:3
777
823
868
9\4
959
005
050
096
141
186
232
277
322
367
412
457
502
547
592
637
682
726
771
816
860
905
949
994
038
083
127
171
216
782
827
873
918
964
009
055
100
146
191
236
281
327
372
417
462
507
552
597
641
686
731
776
820
865
909
954
998
043
087
131
176
220
300 ! 304
348
308
352
392
436
480
524
568
612
656
699
743
787
830
874
917
961
004
396
441
484
528
572
616
660
704
747
791
835
878
922
965
009
344
I
388
432
476
520
564
607
651
695
739
782
826
870
913
957
00 000
~L.
0
I
I
I
:3
3
4
5
I
I
313 i 317
357 I 361
401
445
489
533
577
621
664
708
752
795
839
883
926
970
013
I
405
449
493
537
581
625
669
712
756
800
843
887
930
974
017
322
366
410
454
498
542
585
629
673
717
760
804
848
891
935
978
022
6
I
7
I
8
542
542
542
i 326
370'
414
458
502
546
590
634
677
721
765
808
852
896
939
983
026
!
330
374
419
463
506
550
594
638
682
726
769
813
856
900
944
987
030
i 335
I
9
I
588
588
589
8
: 339
I
9
204\'= 9660" S. 4.68
2 42 = 9720
2 43 = 9780
2 44 = 9840
2 45 = 9900
2 46 = 9960
2 47 = 10020
Prop. Parts
1
2
3
4
5
6
7
8
9
379 i 383
423
427
471
467
511
515
555 559
599
603
647
642
686
691
734
730
774
778
822
817
865
861
904
909
948
952
991 996
035 039 .
~I7 I
3
0015'= 900" S. 4.68557 T. 4.68 558
0 16 = 960
558
557
0 17 = 1020
557
558
2 38 =9480
2 39 =9540
2 40 =9600
I
786 791 795 800 804
809 813
859
832 836 841 845 850 855
877 882 886 891 896 900 905
923 928 932 937 941
946 950
968 973 978 982 987
991 996
023 028 032 037
014 0\9
041
059 064 068 073 078
082 087
127 132
105 109 114 118 123
150 155 159 164 168 173
177
195 200 204 209 214 218
223
241 245 250 254 259 263 268
304 308 313
286 290 295 299
331 336 340 345 349 354 358
376 381 385 390 394
399 403
448
421 426 430 435 439 444
493
466 471 475 480 484 489
534 538
511 516 520 525 529
574 579 583
556 561 565 570
619 623
628
601 605 610 6\4
646 650 655 659 664 668
673
717
691 695 700 704 709 713
758 762
735 740 744 749 753
807
780 784 789 793 798 802
851
825 829 834 838 843 847
892 896
869 874 878 883 887
941
914 918 923 927 932 936
985
958 963 967 972 976 981
*003 *007 *012 *016 *021 *025 *029
069 074
047 052 056 061 065
118
100 105 109 114
092 096
162
136 140 145 149 154 158
180 185 189 193 198 202 207
247 251
224 229 233 238 242
I
5
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
4 ii, 6
5 i 2.0
6 2.4
7 2.8
8 3.2
9 3.6
Prop. Parts
542 T. 4.68589
541
590
541
590
590
541
541
591
541
591
592
540
53
~
T ABLE
N.
L.
1
1000
1001
1002
1003
1004
1005
1006
]007
1008
1009
1010
1011
1012
1013
]014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
]029
1030
1031
1032
-
.-
000 0000
4341
8677
001 3009
7337
002 1661
5980
003 0295
4605
8912
004 3214
7512
005 1805
6094
006 0380
4660
8937
007 3210
7478
008 1742
6002
009 0257
4509
8756
010 3000
7239
011 1474
5704
9931
012 4154
8372
013 2587
6797
QI4 t063
5205
9403015 3598
7788
016 1974
6155
017 0333
4507
8677
018 2843
7005
019 1163
5317
9467
020 3613
7755
021 1893
1633
1034
1035
]036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
N.
.,
0
i
2° 46'=
2 47 =
2 48 =
2 49 =
2 50 =
L.
9960"
10 020
10 080
10140
10200
0
0434
0869
1303
1737
4775
5208
5642
6076
9111
9544
9977 *0411
3442
3875
4308
4741
7770
8202
8635
9067
2093
2525
2957
3389
6411 6843
7275
7706
0726
1157
1588 2019
5036
5467
5898
6328
9342 9772 *020: I *0633
3644 4074
4504
4933
7941 8371
8800
9229
2234
2663
3092
3521
6523 6952
7380
7809
0808
1236
1664 2092
5088 5516
5944
6372
9365 9792 *0219 *0647
363740644490491753445771
7904 8331
8757
9184
2168
2594
3020
3446
6427
6853
7279
7704
0683
1108
1533
1959
4934 5359
5784
6208
9181 9605 *0030 *0454
3424 3848 4272
4696
7662 8086
8510
8933
1897 2320
2743
3166
6127 6550
6973
7396
*0354 *0776 *1198 *1621
4576 4998
5420
5842
8794 9215
9637 *0059
3008 3429
3850
4271
7218
7639 8059
8480
1424 -t844
2264
2685
5625 6045 i 6465 I 6885
9823- *0243 .*066r*I()82
4017 4436
4855
5274
8206
8625
9044
9462
2392
2810
3229
3647
6573
6991
7409
7827
0751
1168
1586 2003
4924
5342
5759
6176
9094
9511 9927 *0344
3259
3676 4092
4508
7421 7837 8253
8669
1578 1994 2410
2825
5732
6147 6562
6977
9882 *0296 *0711 *1126
4027
4442
4856
5270
8169 8583
8997
9411
2307 2720
3134
3547
t
N.
9,
2171
2605
6510
6943
*0844 *1277
5174
5607
9499
9932
3821
4253
8138
8569
2451
2882
6759
7190
* I 063
1493
*
5363
5793
9659 *0088
3950 4379
8238
8666
2521
2949
6799
7227
*1074 *1501
9610
3872
8130
2384
6633
*0878
5120
9357
3590
7818
*2043
6264
*0480
4692
8901
3105
*0037
4298
8556
2809
7058
*1303
5544
9780
4013
8241
*2465
6685
*0901
5113
9321
3525
7305 I 7725
*j50( '*1920
5693
6112
9881 *0300
4065
4483
8245
8663
2421
2838
6593
7010
*0761 *1177
4925
5341
9084
9500
3240
3656
7392
7807
*1540 *1955
5684
6099
9824 *0238
3961
4374
I
3039
3473
7377
7810
*1710 *2143
6039
6472
*0364 *0796
4685
5116
9001
9432
3313
3744
7620
8051
1924 *2354
*
6223
6652
*0517 *0947
4808
5237
9094
9523
3377 3805
7655
8082
*1928 *2355
61986624
*0463 *0889
4724
5150
8981
9407
3234
3659
7483
7907
*1727 *2151
5967
6391
*0204 *0627
4436
4859
8664 9086
*2887 *3310
7107
7529
*1323 *1744
5534
5955
9742 *0162
m5i 8144
8564
+*23-40 *2759
6531
6950
:
*0718
4901
9080
3256
7427
*1594
5757
9916
4071
8222
*2369
6513
*0652
4787
2°
2
2
2
2
I
39(
82~
*257
690
*122'
5548
9863
4174
8481
*2784
7082
*1376
5666
9951'
4233'
8510(
*2782
7051
*1316
5576 '
98321
4084 :
8332
*2575
681~
*1050
5282
9509
*3732
7951
*2165
6376'I'!
*0583
'
I
; 8984
*3178
7369'
1137 *1555
* 5319 5737
9498
9916
3673 4090
7844 8260
*2010 *2427
6173
6589
*0332 *0747
4486
4902
8637
9052
*2784 *3198
6927 7341
* I066 *1479
5201
5614
I
N.
13
51'=10 260" S. 4.68540 T. 4.68593
539
594'
52 = 10320
539
59~
53 = 10 380
539
59J
54 = 10 440
539
595
55 = 10500
54
2° 55'=
2 56
2
2
2
L.
0
021 1893
6027
022 0157
4284
8406
023 2525
6639
024 0750
4857
8960
025 3059
7154
026 1245
5333
9416
027 3496
7572
028 1644
5713
9777
029 3838
7895
030 1948
5997
031 0043
4085
8123
032 2157
6188
033 0214
4238
8257
034 2273
6285
035 OZ93
-4291
8298
036 2295
6289
037 0279
4265
8248
038 2226
6202
039 0173
4141
8106
040 2066
6023
9977
041 3927
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
I 1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
10:19
1100
1\21341561789
S. 4.68541
T. 4.68591
540
592
540
592
540
592
540
593
1050.1100
'fABLE I
1000-1050
112131415161718
I
L.
10500"
=
10 560
57 = 10 620
58 = 10 680
59 =10740
I
2
9060
8659
2674
6686
I
3075
7087,
0093t 1094
j~~!
8698
I
9098
3094
2695
7087
6688
1076
0678
5062
4663
9044
8646
3022
2624
6599 6996
0967
0570
4934
4538
8898
8502
2858
2462
6814
6419
*0372 *0767
4322
4716
0
S. 4.68539
I
539
538
538
538
I
I
I
I
8
6
7
9
5
4787
5201
5614
4374
3961
9332
9745
8919
8506
8093
3459
3046
3871
2634
2221
7582 7994
7170
6345
6758
*0466 *0878 *1289 *1701 *2113
5817
6228
5405
4582
4994
9928 *0339
9517
9106
8695
3625 4036 4446
2804
3214
8139 8549
7729
6909
7319
*1010 *1419 *1829 *2239 *2649
6335 6744
5926
5107
5516
9200
9609 *0018 *0427 *0836
4107
4515 4924
3289
3698
8600 9008
7375
8192
7783
*1457 * 1865 *2273 *2680 *308~
6757 7165
5535
5942
6350
9609 *0016 1*0423 *0830 * 1237
5306
3679
4086
4492
4899
8964
9371
7745
8152
8558
*1808 *2214 *2620 *3026 1*3432
7084
7489
5867
6272
6678
9922 *0327 *0732 *1138 *1543
4783
5188 5592
4378
3973
8830
9234 9638
8425
8020
2064
2872
3277 3681
2468
6912
6104
6508
7315 7719
*0140 *0544 *0947 *1350 *1754
5785
4979 5382
4173
4576
9007
9409 9812
8201
8604
3433 3835
3031
2226
2629
7052
7453 7855
6248
6650
9462 9864 *0265 *0667 *1068 *1470 *1871
4279
4680
5081
5482 5884
3477 3878
8690 9091 9491 9892
7487
7888 8289
I J495 - -r81J5- 7Z1J61Z696 3U96 1349713897
S4~-1~"o
"~n 0 i -669S-LW98-l 149&! -2898I
9498 9898 *0297 '*0697 1*1097 1*1496 '*1896
5491 5890
5091
3494 3893
4293
4692
9082
9481 9880
8683
7486
7885 8284
3468 3867
2671
3070
1475 1874 2272
7849
6655
7053
745\
5460 5858
6257
9442 9839 *0237 *0635 *1033 *1431 *1829
5407
5804
3419 3817
4612
5009
4214
9776
8982
9379
7393 7791 8188
8585
3745
3348
1364
2554
2951
1761 2158
6917
7313
7709
5331 5727
6124
6520
9294 9690 *0086 *0482 *0878 *1274 *1670
4837
5232
5628
4045
4441
3254 3650
9582
8396
8791
9187
7210 7605
8001
*1162 *1557 *1952 *2347 *2742 *3137 *3532
511]
6295
6690
7084 7479
5900
5506
3
2307
2720
3134
6440
6854
7267
0570
0983
1396
4696
5109 5521
9642
8818
9230
2936
3348
3759
7050
7462
7873
1161 1572
1982
5267
5678
6088
9370
9780 *0190
3468
3878 4288
7972 8382
7563
2472
1654 2063
5741
6150
6558
9824 *0233 *064\
3904
4312
4719
8387
8794
7979
2458
2865
2051
6526
6932
6119
*0183 *0590 *0996
4649 5055
4244
8706
9111
8300
3163
2758
2353
6807
7211
6402
1256
0447
0851
5296
4893
4489
8930
9333
8526
2963
3367
'2560
6993 7396
6590
1019 1422
0617
5042
5444
4640
2
3
T. 4.68595
595
596
596
597
4
3547
7680
1808
5933
*0054
4171
8284
2393
6498
*0600
4697
8791
2881
6967
*1049
5127
9201
3272
7339
*1402
5461
9516
3568
7616
1660
5700
9737
3770
7799
1824
5846
4
3° 0'=10
I
I
5
6
7
800" S. 4.68538
3 1 = 10 860
32=10920
33=10980
3 4 = 11 040
537
537
537
537
8
9
T. 4.68597
598
598
599
599
55
ill
TABLE II.
Base of common (Briggs) logarithms = 10.
Base of natural (Napierian) logarithms (e) = 2.718281828459
Modulus of Com. Logs. = logloe = M = 0.4342944819 . . . .
....
1
logelO= 2.30258509299. . . .
M =
1
logeN = !II x logloN.
loglON = M x logeN.
0
I
2
3
4
6
6
7
8
9
10
11
12
13
14
~16
16
17
18
19
20
21
0.00000000
0.43 429 448
0.86858896
1.3028834;
1. 73 717 793
2.17147241
2.60576689
3.04006137
3.47435586
3.90865034
4. 34 294 482
4.77723930
5.2\ 153378
5.64582826
6.0801227;
6.51441723
6.94871171
7. 38' 300 619
7.81 730 067
8. 25 159 516
8.68 588 964
9.12018412
23 I 9.98 877 308 I!
24110.42306757
26
10.85 736 20;
26 11.29165653
27 11.72595 101
28 12. 16 024 549
29 12.59453998
30 13.02 883 446
31 13. 46 312 894
32
33
34
36
36
37
38
39
40
41
42
43
44
46
46
47
48
49
60
13.89742342
14. 33 171 790
14.76601238
15.20 030 687
15.63 460 13;
16.06889583
16.50319031
16.93748479
17.37177928
17.80607376
18.24 036 824
18.67466272
19.10895720
19.54325169
19.97754617
20.41 18406;
20. 84 613 513
21. 28042961
21.71472 410
....
60
Multiples of
To Convert from Com. to Nat. Logs
I
Multiples of M
~ToConvert from Nat. to Com. Logs
21.71472410
0.00000000
117.43183974
119.73442484
46~
36.84 136 149
029
477
925
373
17
18
19
20
39. 14 394 658
41.44653167
43.74911677
46.05 170 186
67
68
69
70
154.27 320 123
156.57578632 .,:
158.87837142
161.18095651
490
822
21
22
48.35
50.65
71
72
163.48 354 160
165. 78 612 670
67
68
69
70
71
22. 58 331 306
23.01760754
23.45190202
?[
2.30258509 I
4.60517019 I
6.90 775528
9.21034037
11.51 292 546
13.81551056
I
2.
3' i
4
25.62 337 443
26.0576689\
26.49 196340
26.92 625 788
27.36 055 236
27.79 484 684
28.22914132
28.66343581
6
6
7
8
9
10
II
12
13
14
16
16
18.42068074
20.72 326 584
23.02 585 093
25.32843602
27.63 102 112
29.93 360 621
32.23 619 130
29.09
29.53
29.96
30.40
23 . 88 619 650
24.32 049 099
24.75 478 547
25. 18 907 995
.
115 . 12 925
773
202
631
061
22.14901858
~?
60
51
52
53
54
66
56
57
58
59
60
61
62
63
64
66
66
51
52
53
54
66
56
57
58
59
60
61
62
63
64
66
66
,t.~3 non:no
73 I 31.70 349 718
74
76
76
77
78
79
80
81
32.13779166
32.57208614
33.00 638 062
33.44 067 511
33.87 496 959
34.30 926 407
34.74 355 855
35. 17 785 303
82
83
84
86
86
87
88
89
90
9\
92
93
94
96
96
97
98
99
100
35.61214752
36.04 644 200
36.48 073 648
36.91503096
37.34 932 544
37.78361993
38.21 79\ 441
38.65 220 889
39.08 650 337
39.52 079 785
39.95 509 234
40.38 938 682
40.82368130
23
24
26\
26
27
28
29
30
31
32
33
34
36
36
37
38
39
40
41
42
43
44
41 .25 797 578 46
41 .69 227 026 46
42.12656474 47
42.56 085 923 48
42.99515371 49
43.42944819 60
16. II 809 565
34.53 877 63,9
428 695
687 20;
5295945714'
i
55.26204223
TABLE
P
FIVE PLACE LOGARITHMS
140.45769067
142.76027577
145.06286086
147.36544 595
149.66 803 104
151.97061 614
P
83
84
86
86
85. 19 564 844 87
87.49 823 353 88
89.80081 863 89
90
92.10340372
91
94.40598881
92
96.7085739\
99.01 115900 93
94
101.31374409
96
103.61632918
96
105.91891428
97
108.22149937
98
110.52408446
99
112.82666956
115.1292546; 100
OF TRIGONOMETRIC
FUNCTIONS
FORMULAS FOR THE USE OF S. AND T.
(For explanations see page 26.)
(1) For a near zero degrees.
J
= log a"
log tan a = log a"
Ing ~in (Y
74 i 170.39129688
+ s.
+ T.
log cot a = epIlog a" + cplT.
= cpllog tan a.
(2) For a near ninety degrees.
172 . 69 388 197
174. 99 646 707
177.29905216
179.60163725
181 .90 422 23;
184.20680744
186.50 939 253 I
III
(For explanations, see pages 23 to 26.)
Pages 58 to 65 give logarithms of sines and tangents for each 10 seconds from 3° to
7° and of cosines and cotangents for each 10 seconds from 83° to 87°. For values of
functions for angles less than 3° and greater than 87° the S. and T. method may be
uscd.
Pages 66 to 110 give logarithms of sines, tangents, cosines, and cotangents for each
minute from 0° to 90°.
138. 15 5\ 0 558
73 . 168 08871 17«1
I'
57.56 462 732 176
76
59.86721 242
62. 16 979 751 77
64.47 238 260 78
66. Tl 496 770 79
69.07 755 279 80
81
71.38013788
73.68 272 298 82
75.98 530 807
78.28789316
80.59 047 825
82.89 306 33;
122.03700993
124.33 959 502'!
126.64218 OIl
128.94476521
131.24735030
133.54 993 539
135.85252049 L
log cos a = log (900
F.
188.81 197763
log tan a
191.11456272
-
a)"
=
epIlog
(900 - a)"
= epilog cot a.
193. 41 714 781
+ S.
log cot a = log (900 - a)" + T.
+
cpl T.
log (900 - a)" = log cas a + cpl S.
= log cot a + cpl T.
= epIlog tan a + cpl T.
195.71 973 290
198.02 231 800
200.32 490 309
202.62748818
204.93 007 328
207.23 265 837
209.53 524 346
211 .83 782 856
214.1404136;
216.44299874
218.74 558 383
221.04816893
223.35 075 402
225.65 333 911
227.95 592 421
230. 25 850 930
.
57
L. cos
9.99
TABLE III
I'
940
940
939
938
938
937
936
936
935
934
934
933
932
932
931
930
929
929
928
927
926
926
925
924
923
923
922
921
920
920
919
918
917
917
916
915
9\4
913
913
912
911
9\0
909
909
908
907
906
905
904
904
903
902
901
900
899
898
898
897
896
895
0"
0
I
2
3
4
5
6
7
8
9
10
1\
12
13
14
15
16
17
18
\9
20
2\
22
23
24
25
26
27
28
29
30
3\
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
56
56
57
58
59
8.71 880
8. 72 120
359
597
834
8.73 069
303
535
767
997
8.74 226
454
680
906
8.75 130
353
575
795
8.76 015
234
451
667
883
P,.77 097
310
522
733
943
8.78 152
360
568
774
979
8.79
183
10" 20"
920 960
160 200
399 439
637 676
873 912
108 147
342 380
574 613
805 844
*035 *073
264 302
491 529
718 755
943 980
167 204
390 427
612 648
869
832
052 088
270 306
487 523
703 739
919 954
\33
168
346 381
558 593
768 803
978 *013
187 222
395 430
602 636
808 842
*013 1*047
217
386 I 420.
588 622
789 823
990 *023
8.80 189 222
388 421
585 618
782 815
978 *010
8.81 173 205
367 399
560 592
752 784
944 975
8.82 134 166
324 356
513 544
732
70\
888 920
106
8.83 075
261 292
446 476
630 660
813 844
996 *026
JI.84 177 208
J 30" ! 40" I so" l 60"
*000
240
478
716
95\
186
4\9
651
882
*112
340
567
793
*018
241
464
685
905
125
343
559
775
990
204
416
628
838
*048
257
464
671
8761
*08\
284
251
*040
280
518
755
991
225
458
690
920
*150
378
605
831
*055
279
501
722
942
\61
379
595
81\
*026
239
452
663
873
*083
291
499
705
910
*115
I 453 I 487. I
60"
655
856
*056
255
454
65\
847
*043
237
431
624
816
*007
198
387
576
764
951
137
322
507
691
874
*056
238
~40"
log cos
58
log sin
3°
318
*080
320
558
794
*030
264
497
728
959
*188
416
642
868
*092
316
538
759
979
197
115
;31
847
*061
275
487
698
908
*118
326
533
739
9451
*149
35~
521 I. 555
*120
359
597
834
*069
303
535
767
997
*226
454
680
906
*130
353
475
795
*015
234
451
667
883
*097
310
522
733
943
*152
360
568
774
979
*183
386
I
588
689 ! 722 i 756 i 789
i
890
*090
289
487
684
880
*075
270
463
656
848
*039
229
4\9
607
795
982
168
353
538
721
904
*087
268
I
30"
923
*123
322
519
716.
913
*108
302
496
688
880
*071
261
450
639
826
*0\3
199
384
568
752
935
*117
298
956
*156
355
552
749
945
*\40
334
528
720
912
*103
292
482
670
857
*044
230
4\5
599
783
965
*147
328
~IO"
86°
990
*189
388
585
782
978
*173
367
560
752
944
*134
324
5\3
701
888
*075
261
446
630
813
996
*177
358
I
Id
I
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
1\
10
9
8
7
6
5
4
3
2
I
0
940
939
938
938
937
936
936
935
934
934
933
932
932
931
930
929
929
928
927
926
926
925
924
923
923
922
921
920
920
919
918
917
917
916
915
914
913
913
912
911
910
909
909
908
907
906
905
904
904
903
902
901
900
899
898
898
897
896
895
894
40
40
40
40
39
39
39
39
38
38
38
38
38
37
37
37
37
37
36
36
36
36
36
36
35
35
35
35
35
35
34
34
34
34
34
34
34
33
33
33
33
33
32
32
32
32
32
32
32
32
31
31
31
3\
3\
31
30
30
30
30
TABLE III
L.sin
40
4.0
1
39
3.91
2
3
8.0 7.8
12.0 11.71
4 I. 16.0 15.6
5 20.0 19.5'
6 24.0 23.4
7 28.0 27.3
8 32.0 31.2
9 36.0 35.1
38
37
\
3.8 3.7
2
7.6 7.4
3 1\.41\.1
4 15.2 14.8
5 19.0 18.5
6 22.8 22.2
7 26.6 25.9
8 30.4 29.6
9 34.2 33.3
36
3.6
I
7.2
2
3 10.8
4 14.4
5 18.0
6 2\.6
7 25.2
8 28.8
9 32.4
34
35
1
3.5
2
7.0
6.8
10.5
10.2
140
13.6
3
4
I
I
3.4
5 i 17.5 ILl!
6
7
8
9
I
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
~dI
0"
,
, Prop. Parts
2\ . 0
24 . 5
28.0
31 . 5
33
3.3
6.6
9.9
13.2
16.5
19.8
23.1
26.4
29.7
31
3.1
6.2
9.3
12.4
15.5
18.6
21.7
24.8
27.9
20. 4
23. 8
27.2
30. 6
32
3.2
6.4
9.6
\2.8
16.0
19.2
22.4
25.6
28.8
30
3.0
6.0,
9.0
12.0
15.0
18.0
2\.0
24.0
27.0
Prop. Parts
0
I
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
! 34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
5\
52
53
54
55
56
57
58
59
all
I
8.80
8.81
8.82
8.83
8.84
I
300
3°"
40"
*060
301
540
777
*014
249
483
716
947
178
407
634
86\
*087
311
534
756
977
197
416
633
850
*065
280
493
706
917
127
337
545
752
958
164
*100
341
579
817
*053
288
522
754
986
216
445
672
899
*124
348
571
793
*014
233
452
669
886
*101
3\5
529
741
952
162
371
579
787
993
198
368
402
20"
10"
8.71 940
980 *020
8.72 18\
221
261
420 460
500
659 698 738
896 935 975
8.73 132 171 210
366 405 444
600 638 677
832 870 909
8.74 063
101 \39
292 330 369
521 559 597
748 786 823
974 *012 *049
8.75 \99
236 274
423 460 497
645 682 719
867 904 940
8.76 087
\24
160
306 343 379
525 561 597
742 778 8\4
958 994 *030
8.77 \73
208 244
387 422 458
600 635 670
811 847 882
8.78 022 057 092
232 267 302
441 475 5\0
649 683 718
855 890 924
130
8. 79 061 096
266
I
334
I
470 i 504 i 538 i 572 606
673 707 741 774 808
875 909 942 976 *009
076
110 143 177 210
277 310 343 376 409
476
509 542 575 608
674 707 740 773 806
872 905 937
970 *003
068
101 134 166 199
264 297 329 362 394
459 491 524 556 588
653 685 717 750 782
846 878 910 942 974
038 070
102 134 166
230 262 293
325 357
420 452 484 515 547
610 642 673 705 736
799 831 862
893 925
987 *0\9 *050 *081 *\12
175 206 237 268 299
361
392 423
454 485
547 578 609
640 671
732 763 794 824 855
9\6
947 978 *008 *039
100 130 16\
191 222
282 313 343 374 404
6011
log tan
3°
50"
I 4°"
~wt.
30"
20°
5;)11
*141
380
619
856
*093
327
561
793
*024
254
483
710
936
*\62
385
608
830
*051
270
488
706
922
*137
351
564
776
987
197
406
614
821
*027
232
I
I
436
639
I
*181
420
659
896
*\32
366
600
832
*063
292
521
748
974
*199
423
645
867
*087
306
525
742
958
*\73
387
600
811
*022
232
441
649
855
*061
266
I
I
470
673
842
*043
243
443
641
839
*036
232
427
621
814
*006
198
389
579
768
956
*144
330
5\6
701
886
*069
252
434
875
*076
277
476
674
872
*068
264
459
653
846
*038
230
420
610
799
987
*\75
361
547
732
916
*100
282
464
10"
0"
- --86°
59
58
57
56
55
54
53
52
5\
50
49
48
47
46
45
44
43
42
4\
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
I
0
,
40
40
40
40
39
39
39
39
38
38
38
38
38
38
37
37
37
37
36
36
36
36
36
36
36
35
35
35
35
35
34
34
34
34
34
34
34
34
33
33
33
33
33
32
32
32
32
32
32
32
32
31
31
31
31
31
31
31
30
30
I
Prop. Parts
J
d
6.)"
d
41
4.\
8.2
12.3
16.4
20.5
24.6
28.7
32.8
36.9
39
3.9
I
2
7.8
3 11.7
4 \5.6
5 19.5
6 23.4
7 27.3
8 31.2
9 35. I
37
I
3.7
2
7.4
3 I\. I
4 14.8
5 18.5
6 22.2
7 25.9
8 29.6
9 33.3
35
3.5
11 7.0
1
2
3
4
5
6
7
8
9
3
4
5
6
7
8
9
I
2
3
4
5
6
7
8
9
I
2
3
4
5
6
7
8
9
I
I
40
4.0
8.0
12.0
\6.0
20.0
24.0
28.0
32.0
36.0
38
3.8
7.6
I\.4
15.2
19.0
22.8
26.6
30.4
34.2
36
3.6
7.2
10.8
14.4
18.0
2\.6
25.2
28.8
32.4
34
3.4
6.8
10.2
10.5
14.0 13.6
17.5 17.0
21.0 20.4
24.5 23.8
28.0 27.2
31.5 30.6
32
33
3.2
3.3
6.4
6.6
9.6
9.9
13.2 12.8
16.5 16.0
19.8 19.2
23.1 22.4
26.4 25.6
29.7 28.8
30
31
3.0
3. I
6.0
6.2
9.0
9.3
12.0
12.4
15.5 15.0
18.0
18.6
21.7 21.0
24.8 24.0
27.9 27.0
Prop. Parts
n______----
59
L. cos
TABLE III
r--'9.99
894
893
892
891
891
890
889
888
887
886
885
884
883
882
881
880
879
879
878
877
876
875
874
873
872
871
870
869
868
867
866
865
864
863
862
861
860
859
858
857
856
855
854
853
852
851
850
848
847
846
845
844
843
842
841
840
839
838
837
836
40
0"
0 8.84
1
2
3
4 8.85
5
6
7
8
9
10 8.86
11
12
13
14
15
16 8.87
17
18
19
20
21
22 8.88
23
24
25
26
.
27
28 8.89
29
30
31
32
33
34 8.90
35
36
37
38
39
40 8.91
41
42
43
44
45
46
47 8.92
48
49
50
51
52
53 8.93
54
55
56
57
58
59
I
60
I
358
539
718
897
075
252
42~
605
780
955
128
301
474
645
816
987
156
325
494
661
829
995
161
326
490
654
817
980
142
304
464
625
784
943
102
260
417
574
730
885
040
195
349
502
655
807
959
110
261
411
561
710
859
007
154
301
448
594
740
885
60"
10"
20"
389
569
748
927
105
282
458
634
809
984
157
330
502
674
845
*015
185
354
522
689
856
*023
188
353
518
681
845
*007
169
330
491
651
811
419
599
778
957
134
311
488
663
838
*013
186
359
531
703
873
*043
213
382
550
717
884
*050
216
381
545
709
872
*034
196
357
518
678
837
"
970
I
996
I
449
629
808
986
164
341
517
693
867
*042
215
388
560
731
902
*072
241
410
578
745
912
*078
243
408
572
736
899
*061
223
384
545
704
864
40"
479
659
838
*016
193
370
546
722
896
*070
244
416
588
760
930
*100
269
438
606
773
940
*106
271
436
600
763
926
*088
250
411
571
731
I 890
5°"
509
688
867
*045
223
400
576
751
926
*099
273
445
617
788
958
*128
297
466
634
801
967
'33
198
463
627
790
953
* 115
277
438
598
758
917
*023
1*049
*075
30"
TABLE
log sin
~
128 I 154 i 181 I 207 ' 233
286 312 338 364 391
443 469 495 521 548
600 626 652 678
704
756 782 808 834
859
911 937 963
989 *015
066 092
118
143 169
221 246 272 298
323
374 400 426 451
477
528 553 579 604
630
680 706 731 757
782
833 858 883 909 934
984 *010 *035 *060 *085
135 161 186 211 236
286 311 336 361 386
436 461 486
511 536
586 611 636 660 685
735 760 784 809 834
883 908 933 957 982
031 056 081
105 130
179 203 228 253 277
326 350 375 399 424
472 497 521 546 570
619 643 667 691 716
764 788 812
837 861
909 933: 957
981 *006
60"
1
I
I
I
539 59 893
718 58 892
897 57 891
*075
56 891
252 55 890
42~ 54 889
605 53 888
78Q 52 887
955
51 886
* 128 50 885
301 49 88~
474 48 883
645 47 882
816 46 881
987 45 880
*156
44 879
325 43 879
494 42 878
661 41 877
829 40 876
995
39 875
*161
38 874
326 37 873
490
36 872
654 35 871
817 34 870
980
33 869
* 142 32 868
304
31 867
464 30 866
625 29 865
784 28 864
943 27 863
1*102
I
Prop. Parts
d
30
30
30
30
30
30
29
29
29
29
29
29
29
29
28
28
28
28
28
28
28
28
28
27
27
27
27
27
27
27
27
26
26
260
26
25
862
26
417
574
730
885
*040
195
349
502
655
807
959
* 110
261
411
561
710
859
*007
154
301
448
594
740
885
*030
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
860
859
558
857
856
855
854
853
852
851
850
848
847
846
845
844
843
842
841
840
839
838
837
836
834
26
26
26
26
26
26
26
26
26
25
25
25
25
25
25
25
25
25
24
24
24
24
24
24
24
d
I 50" I 40" I 30" I 20" I 10" I 0" I
'
861
1
~
4
5
6
~
9
29
2.9
5.8
8.7
11.6
14.5
17.4
20.3
23.2
26.1
1
2
3
4
5
6
7
8
9
26
I 9.99 I
i
26
2.6
5.2
7.8
10.4
13.0
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
15.6
18.2
20.8
23.4
0"
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
8.84 464
646
826
8.85 006
185
363
540
717
893
8.86 069
243
417
591
763
935
8.87 106
277
447
616
785
953
8.88 120
287
453
618
783
948
8.89 111
274
437
598
760
920
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
399
557
715
872
8.91 029
185
340
495
650
803
957
8.92 110
262
414
565
716
866
8.93 016
165
313
462
609
756
903
8.94 049
60"
,4
40
III
8.90 080
240
10"
~O"
\ 30"
log tan
50"
40"
555 585 615
736 766 796
916 946 976
125 155
095
274 304 333
452 481 511
629 658 688
805 835 864
981 '010 1'039
156 185 214
330 359 388
504 533 562
677 706 734
849 878 907
*021 '049 *078
192 I 220 249
3621 390 419
532 560 588
701 729 757
869 897 925
*037 "065 *092
204 231 259
370 398 425
536 563 591
701 728 756
866 893 920
*029 *057 *084
193 220 247
355
383 410
545 571
518
679 706 733
894
840 1867
*000 *027 1*054
134 160 I 187 I 213
293 I 319 i 346 I 372
451' I 478 1 504 1 531
495
525
676
706
886
856
036 065
214 244
392 422
570 599
776
747
922 952
127
098
272 301
447 475
648
619
792 821
992
964
135 163
305 334
475 503
644 673
813 841
981 *009
148 176
315 342
508
481
646 674
811 838
975 *002
138 166
301 328
464 491
625 652
786 813
947 974
107
I 26t. I
425
610 636 ' 662 688
583
793 820 846
741 767
898 924 950 976 *002
107 133 159
081
055
262 288 314
211 236
443
469
418
366 392
572
598 624
521 547
727
752 778
675 701
880 906 931
829 855
982 *008 *033 *059 *084
186 211 237
160
135
338 363 388
313
287
515 540
464 489
439
640 665 691
590 615
766 791 816 841
741
891
916 941 966 991
115 140
040
065 090
190 214 239 264 289
388 412 437
363
338
585
536 560
486
511
732
658 6831707
634
805 830 854 879
781
952 976 *001 *025
928
122 147 171
098
074
I
I
50"
30"
4°"
log cot
I
20"
10"
- --
86°
I 60"
646
826
*006
185
363
540
717
893
*069
243
417
591
763
935
*106
277
447
616
785
953
*120
287
453
618
71\3
948
*111
274
437
598
760
920
*080
240
399
557
715
872
*029
185
340
495
650
803
957
*110
262
414
565
716
866
*016
165
313
462
609
756
903
*049
195
0"
Prop. Parts
d
30
30
30
30
30
30
30
29
29
29
29
29
29
29
28
28
28
28
28
28
28
28
28
28
28
28
27
27
27
27
27
27
27
27
26
26
26
26
26
26
26
26
26
26
26
26
25
25
25
25
25
25
25
25
25
24
24
24
24
24
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
I
I
I
d
31
3.1
6.2
9.3
12.4
15.5
18.6
21.7
24.8
1
2
3
4
5
6
7
8
9
I
30
3.0
6.0
9.0
12.0
15.0
18.0
21.0
24.0
27.0
27.9
29
2.9
5.8
8.7
11.6
14.5
17.4
20.3
23.2
26.1
1
2
3
4
5
6
7
8
9
28
2.8
1
5.6
2
8.4
3
4 11.2
5 14.0
6 16.8
7 19.6
8 22.4
9 ,25.2
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
27
2.7
5.4
8.1
10.8
13.5
16.2
18.9
21.6
24.3
26
2.6
5.2
7.8
10.4
13.0
15.6
18.2
20.8
23.4
25
2.5
5.0
7.5.
10.0
12.5
15.0
17.5
20.0
22.5
24
2.4
4.8
7.2
9.6
12.0
14.4
16.8
19.2
21.6
Prop. Parts
L. cos
Table
9.99
834
833
832
831
830
829
828
827
825
824
823
822
821
820
819
817
816
815
814
813
812
810
809
808
807
806
804
803
802
801
800
798
797
796
795
793
792
791
790
788
787
786
785
783
782
781
780
778
777
776
775
773
772
771
769
768
767
765
764
763
0"
0
1
2
3
4
5
6
7
8
9
10
II
12
13
14
15
16
17
18
19
20
21
.22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
10"
I 20"
8.94 030 054' 078
174 198 222
317 341 365
461 484 508
603 627 651
746
769 793
887 911 935
8.95 029 052 076
170
193 216
310
333 357
450
473 496
589 613 636
728 752 775
867
890 913
8.96005
028 051
143 166 189
280
303 326
41 7 440 462
553 576 599
689
712 735
825
847 870
960 982 *005
8.97 095
117 139
229 251 274
363
385 407
496 518 541
629 651 674
762
784 806
894 916 938
8.98 026 048
070
157 179 201
288 310 332
419 .~41 462
549 571 592
679
701 722
808 ! 830 ] 851
937 959 980
8.99066
087
109
194 216 237
322 343 365
450 471 492
577 598 619
704 725 746
830 851 872
956 977 998
9.00 082
103 123
207 228 249
332 353 373
456
477 498
581
601 622
704
725 746
828 848 869
951 971 992
9.01 074 094
115
196 217 237
318 339 359
440 460 480
561 582 602
682 703 723
803 823 843
I
60"
log cos
50
III
126
270
413
556
698
840
982
123
263
403
543
682
821
959
097
234
371
508
644
780
915
*050
184
318
452
585
718
850
982
114
245
375
506
636
765
i 894
*023
152
280
407
534
661
788
914
*040
165
290
415
539
663
787
910
*033
155
278
399
521
642
763
883
30"
20"
I
I
84°
1.50
294
437
580
722
864
*005
146
287
427
566
705
844
982
120
257
394
531
667
802
937
*072
207
341
474
'{)7
140
872
*004
135
266
397
527
657
787
916
*045
173
301
428
556
682
809
935
*061
186
311
436
560
684
807
930
*053
176
298
420
541
662
783
903
J
10"
I
60"
40" I 50"
3°"
102
246
389
532
675
817
958
099
240
380
520
659
798
936
074
212
349
485
621
757
892
*027
162
296
430
563
696
828
960
092'
223
354
484
614
744
]
873
002
130
258
386
513
640
767
893
*019
144
269
394
518
642
766
889
*012
135
257
379
501
622
743
863
174
317
461
603
746
887
*029
170
310
450
589
728
867
*005
143
280
417
553
689
825
960
*095
229
363
496
629
762
894
*026
157
288
419
549
I 679
I
808
j 937
*066
194
322
450
577
704
830
956
*082
207
332
456
581
704
828
951
*074
196
318
440
561
682
803
923
1
0"
5°
TABLE III
log sin
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
833
832
831
830
829
828
827
825
824
823
822
821
820
819
817
816
815
814
813
812
810
809
808
807
806
804
803
802
801
800
798
797
796
795
793
792
791
790
788
787
786
785
783
782
781
780
778
777
776
775
773
772
771
769
768
767
765
764
763
761
'~~
L. sin
d
24
24
24
24
24
24
24
24
23
23
23
23
23
23
23
23
23
23
23
23
22
22
22
22
22
22
22
22
22
22
22
22
22
22
22
22
22
21
21
21
21
21
21
21
21
21
21
21
21
21
21
20
20
20
20
20
20
20
20
20
0"
I
1
2
3
4
5
6
7
8
9
24
2.4
4.8
7.2
9.6
12.0
14.4
16.8
19.2
21.6
1
2
3
4
5
6
7
8
9
23
2.3
4.6
6.9
9.2
11.5
13.8
16.1
18.4
20.7
1
2
3
4
5
6
7
8
9
22
2.2
4.4
6.6
8.8
11.0
13.2
15.4
17.6
19.8
I
1
2
3
4
5
6
7
8
9
21
2.1
4.2
6.3
8.4
10.5
12.6
14.7
16.8
18.9
1
2
3
4
5
6
7
8
9
20
2.0
4.0
6:0
8.0
10.0
12.0
14.0
16.0
18.0
Prop. Parts
0
I
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
i 35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
195
340
485
630
773
917
8.95 060
202
344
486
627
767
908
8.96047
187
325
464
602
739
877
8.97 013
150
285
421
556
691
825
959
8.98 092
225
358
490
622
753
10"
8.94
88i
8.99 015
I
I
20"
I 3°"
I 50"
340
485
630
773
917
*060
202
344
486
627
767
908
*047
187
325
464
602
739
877
*013
150
285
421
556
691
825
959
*092
225
358
490
6Z2
753
884
950
I
971
037 i 058 i 080
I
102 i 123 i 145
906
I
928
061
\ 50"
081
1
40"
' 232
361
491
620
748
876
*004
131
258
385
511
637
763
888
*013
138
262
386
509
632
755
878
*000
101 121
I ~o"
log cot
I
993
' 253
383
512
641
769
898
*025
153
280
406
532
658
784
909
*034
158
282
406
530
653
776
898
*020
I 20" I
142
JO"
-
log tan
I 60"
219
244 268 292 316
365
389 413 437 461
509 533 557 581 606
654
678 702 725 749
797 821
845 869 893
941 964 988 *012 *036
083
107 131 155 178
226
249 273 297 320
368 391 415 439 462
509 533 556 580 603
650
674 697 721 744
791 814 838 861 884
931
954 977 *001 *024
071
094
117 140 163
210 233 256 279 302
349
372 395 418 441
487 510 533 556 579
625 648 671 694 717
762 785 808 831 854
899 922 945 968 991
036 059 081
104 127
172 195 218 240 263
308 331 353 376 398
443
466 488 511 533
578 601 623 646 668
713
735 758 780 802
847 869 892 914 936
981 *003 *025 *048 *070
114
136 159 181 203
247 269 291 314 336
380 402 424 446 468
512 534
556 578 600
644 666 687 709 731
775
797 819 841 862
145 ' 167 ' 188 ' 210
275 297
318 340
405 426 448 469
534 555
577 598
662 684
705 727
791 812 834 855
919 940 961 983
9.00 046 068
089
110
174
195 216 237
301 322
343 364
427 448
469 490
553 574
595 616
679
700 721
742
805 826 846 867
930 951 971 992
9.01 055 075 096
117
179 200 220
241
303 324 344
365
427 447 468 489
550 571
591 612
673 694
714 735
796 816 837 857
918 939 959 979
Q
02 040
60"
I 40"
*015
' 275
405
534
662
791
919
*046
174
301
427
553
679
805
930
*055
179
303
427
550
673
796
918
*040
1
162
I
0"
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
80
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
d
24
24
24
24
24
24
24
24
24
24
23
23
23
23
23
23
23
23
23
23
23
23
23
22
22
22
22
22
22
22
22
22
22
22
22
22
22
22
22
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
20
20
20
20
20
20
d
Prop. Parts
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
I
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
I
.
1
I
25
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
22.5
24
2.4
4.8
7.2
9.6
12.0
14.4
16.8
19.2
21.6
23
2.3
4.6
6.9
9.2
11.5
13.8
16.1
18.4
20.7
22
2.2
4.4
6.6
8.8
11. 0
13.2
15.4
17.6
19.8
21
2. I
4.2
6.3
8.4
10.5
12.6
14.7
16.8
18.9
20
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
18.0
Prop. Parts
L.cos
9.99
761
760
759
757
756
755
753
752
751
749
748
747
745
744
742
741
740
738
737
736
734
733
731
730
728
727
726
724
723
721
720
718
717
716
714
713
711
710
708
707
705
704
702
701
699
698
696
695
693
692
690
689
687
686
684
683
681
680
678
677
'
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
66
56
57
58
59
TABLE IIT
10" I 20" I 3°"
6"
943 964
9.01 923
984
083
9.02 043 063
103
163 183 203 223
302 322 342
283
402 421 441 461
540 560 579
520
639
658 678 698
757 776 796 816
874
894 914 933
992 *011 *031 *050
9.03 109 128 148 167
226 245 265 284
342
361 381 400
458 478 497 516
574 593 613 632
690 709 728 747
824 843 862
805
920 939 958 977
053 072 091
9.04 034
149 168 187 206
281 300 319
262
376 395 414 433
490
508 527 546
603 621 640 659
734 753 772
715
828 847 865 884
940 959 977 996
9.05 052 071 089
108
182 201 219
164
275 293 312 330
386 404 423 441
497
515 533 552
607 625 644 662
717 736 754! 772
827 845 864 i 882
937 i 955 973 i 991
9.06 046 064 082' I 0 I
155 173 191 210
264 282 300 318
372 390 408 426
481 499 517 535
606 624 642
589
696 714 732 750
804 821 839 857
911 929 946 964
035 053 071
9.07 018
124 142 160 177
231 248 266 284
337 354 372 390
442 460 478 495
548 566 583 601
653 671 688 706
758 776 793 811
881 898
915
863
968 985 *002 *020
107 124
9.08 072 089
176 193 211 228
280 297 314
331
383 400 418 435
486 504
521 538
I 50"
60"
30"
4°"
log cos
.
fr'
4°"
*004
123
243
362
481
599
717
835
953
*070
187
303
420
535
651
766
881
996
110
225
338
452
565
678
790
903
*015
12(,
238
349
460
570
681
791
I 900
i
*043
163
283
402
520
639
757
874
992
109
*
226
342
458
574
690
805
920
*034
149
262
376
490
603
715
828
940
*052
164
275
386
497
607
717
827
937
I
264
228 246
336
354 372
445 463
481
553 571 51\9
660 678
696
768
786 804
875 893 911
982 *000 *018
089
106 124
195 213 231
301 319 337
407 425 442
513 530 548
618 636
653
723 741 758
828 846
863
933 950
968
*037 *055 *072
141 159 176
245 262
280
349 366
383
486
452 469
555 572 589
83°
10"
d
60"
,*010 ,*028
,*046
I
' 119 I 37 155
20"
TABLE III
log sin
j 50"
*024
143
263
382
501
619
737
855
972
*089
206
323
439
555
670
786
901
*015
129
244
357
471
584
697
809
921
*033
145
256
367
478
589
699
I 809
I 918
0"
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
6
4
3
2
1
0
,
760
759
757
756
755
753
752
751
749
748
747
745
744
742
741
740
738
737
736
734
733
731
730
728
727
726
724
723
721
720
718
717
716
714
713
711
710
708
707
705
704
702
701
699
698
696
695
693
692
690
689
687
686
684
683
681
680
678
677
675
9.99
20
20
20
20
20
20
20
20
20
20
20
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
17
17
17
17
17
17
d
Prop. Parta
1
2
3
4
5
6
7
8
9
21
2.1
4.2
6.3
8.4
10.5
12.6
14.7
16.8
18.9
1
2
3
4
5
6
7
8
9
20
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
18.0
1
2
3
4
5
6
7
8
9
19
1.9
3.8
5.7
7.6
9.5
11.4
13.3
15.2
17.
'
.
i
i
1
2
3
4
5
6
7
8
9
18
1.8
3.6
5.4
7.2
9.0
10.8
12.6
14.4
16.2
1
2
3
4
5
6
7
8
9
17
1.7
3.4
5.1
6.8
8.5
10.2
11.9
13.6
15.3
Prop. Parts
L.mn
A
0
I
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
66
56
57
58
59
I
0"
6°
10"
20"
9.02
30"
162
182 203
223
283
304
324 344
404
425 445
465
525
545 565
585
645
706
666 686
766
785 805 825
885 905
925 945
9.03 005 025 044 064
124
144 163 183
242 262 282
302
361 381 400 420
479 499
518 538
597
616 636 656
714 734 753 773
832
851 871 890
ge7 *007
948 968
9.04 065
084
104 123
181 201
220 239
297
317 336 355
413 432 451 471
528
548 567
586
643 663 682
701
758
777 796 815
873 892 911 930
987 *006 *025 *044
9.05 101 120 139 158
233
214
252 271
328 347
365 384
441 460 478 497
572 591 610
553
666 685
703 722
797 815 834
778
890 909
927 946
9.06 002 a20
039! 057
113 132 I 150 I 169
224 243 261 i 279
335 I 353
372 . 390
500
445 464 482
556 574
592 611
666
684 702 720
775 793 812 830
939
885 903 921
994 *012 *030 *048
9.07 103 121 139 157
211 229 247 266
320 338
356 374
428
446 464 482
536 554 572 589
643 661
679 697
751 768 786 804
858 875 893 911
964 982 *000 *018
9.08 071
089
106 124
177 195 213 230
283 301 319
336
389 407
424 442
495 512
530 .547
600 617
635 653
705 722 740
757
810
827 I 845
862
;;0" 40"
60"
3°"
Jog cot
, ,.
I
I 4°"
243
364
485
605
726
845
965
084
203
321
440
558
675
793
910
*026
143
259
374
490
605
720
835
949
*063
177
290
403
516
628
741
853
964
076
I 187
i 298
i
409
519
629
739
848
957
*066
175
284
392
500
607
715
822
929
*035
142
248
354
460
565
670
775
880
50"
263
384
505
625
746
865
985
104
223
341
459
577
695
812
929
*046
162
278
394
509
624
739
854
968
*082
195
309
422
535
647
759
871
983
094!
206
i 316
I
. 537
427
647
757
866
976
*085
193
302
410
518
625
733
840
947
*053
160
266
371
477
582
688
792
897
j
\ 20" 10"!
83°
~ogtan
j
I
60"
283
404
525
645
766
885
*005
124
242
361
479
597
714
832
948
*065
181
297
413
5211
643
758
873
987
*101
214
328
441
553
666
778
890
1*002
113
j 224
i 335
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
26
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8.
7
6
5
4
3
2
1
0
: 445
556
666
775
885
994
*103
211
320
428
536
643
751
858
964
*071
177
283
389
495
600
705
810
914
0'
I
'
d
I
20
20
20
20
20
20
20
20
20
20
20
20
20
20
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
17
I
d
Prop. Parts
1
2
3
4
5
6
7
8
9
21
2.1
4.2
6.3
8.4
10.5
12.6
14.7
16.8
18.9
1
2
3
4
5
6
7
8
9
20
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
18.0
19
1
1.9
3.8
2
3
5.7
4
7.6
5
9.5
6 11.4
7 13.3
8 ! 15.2
9 I 17. I
I
1
2
3
4
5
6
7
8
9
18
1.8
3.6
5.4
7.2
9.0
10.8
12.6
14.4
16.2
1
2
3
4
5
6
7
8
9
17
1.7
3.4
5.1
6.8
8.5
10.2
11.9
13.6
15.3
Prop. Parts
~
TABLE III
0°
log sin
6~ ?
6:46'373
120 2 6.76476
180 3 6.94 08;
240 4 7.06579
300 5 7.16270
360 6 7.24 188
420 7 7.30882
480 8 7. 36 682
540 9 7.41797
600 0 7.46373
66011 7.50512
72012 7.54291
78013 7.57767
840 14 7.60 985
900 15 7.63 982
96016 7.66784
1020 17 7.69 417
] 080 18 7. 71 900
114019 7.74248
1200 0 7. 76 475
126021 7.78591
132022 7.80 615
138023 7.82545
144024 7.84393
1500 5 7.86 166
156026 7.87 870
162027 7.89509
1680 28 7.91 088
174029 7.92 612
1800 0 7.94 084
1860 31 7.95508
192032 7.96887
198033 7.98223
2040 34 7.99 520
2100
216036
222037
228038
234039
2400 0
246041
252042
258043
8.02002
8.03192
8.04 350
8.05478
8.06 5Z8
8.07650
8.08 696
8.09 718
264044
2700 5
276046
282047
2880 48
2940 49
3000
3060 51
312052
3180 53
3240 54
3300 5
336056
342057
3480 58
354059
3600
8. 10 717
8. 11 693
8.12647
8.13 581
8. 14 495
~. 15 391
8. 16 268
8. 17 128
8.17971
8. 18 798
8. 19 610
8.20 407
8.21189
8.21958
8.22 713
8.23 456
8.24186
cpl S
d
I
cpl T
3010315:3j'443
17609! 5.31443
12494 5.31 443
9691 5.31443
7918 5.31443
6694 5.31 44?
5800 5.31443
5115 5 . 31 443
4576 5.31443
4139 5.31443
3779 5.31443
3476 5.31443
3218 5.31443
2997 5.31 443
2802 5.31 443
2633 5.31443
2483 5.31 443
2348 5.31 443
2227 5.31443"
2119 5 . 31 443
2021 5.31443
1930 5.31443
1848 5.31443
1773 5.31443
1704' 5.31 443
1639 5.31 443
1579 5.31443
1524 5.31 443
1472 5.31443
1424 5.31 443
1379 5.31443
1336 5.31443 j
1297 5.31443
1
5.31
1259
443
.
11901:531443
1158 5.31443
1128 5.31 443
1100 5.31443
1072 5.31443
1046 5.31444
1022 5.31 444
999 5.31 444
5.31 443
5.31443
5.31443
5.31442
5.31442
5. 31 442
5.31 442
5.31 442
5.31442
5.31442
5.31442
5.31442
5.31442
5.31442
5.31442
5.31442
5.31442
5.31 442
5.31442
5.31442
5.31442
5.31 442
5.31 442
5.31442
5.31 442
5.31442
5.31442
5.31442
5.31441
5.31441
5.31 441
5.31 441
d
log cot
cd
log cos
''''''''
6.46373
6.7647Q
6.94085
7.06579
7.16270
30103 3.53627
17609 3.23524
12494 3.05915
9691 2.93421
2.83730
7. 24 188 7918 2. 75 812
6694
7.30 882
2.69 118
7.36 682 5800 2.63 318
5115
7.41797
4576 2.58203
7.46373
4139 2.53627
7.50512
3779 2.49488
7.54291
3476 2.45709
7.57767
3219 2.42233
7.60986
2996 2.39014
7.63 98~
2803 2.36018
7.66785
2633 2.33215
7.69418
2.30582
7.71 900 2482 2.28 100
2348
7.74248
2228 2.25752
7.76476
2119 2.23521
7.78595
2020 2.21402
7.80 615 1931 2.19 385
7.82 546
2. 17 454
7.84394 1848
17732.15606
7.86 167
7.87871
7.89510
7.91089
7.92613
7.94086
7 95 510
7 J6 889
'
1704 2. 13 833
2.12129
1639
157912.10 490
1524 2.08911
1473 2.07387
1424 2.05914
1379 2.04 490
1336 2.03 111
1
/
0.00000
0.00 000
0.00 000
0.00000
0.00 000
0.00000
O.00 000 54
0.00 000 53
0.00 000 52
0.00 000 51
0.00000 60
0.00000 49
0.00 000 48
0.00 000 47
0.00 000 46
0.00000 46
0.00 000 44.
9.99999 43
9.99 999 42
9.99999 41
9.9999940
9.99999 39
9.99 999 38
9.99 999 37
9.9999936
9.99 999 36
9.99999 34
9.99999 33
9.99999 32
9.99998 31
9.9999830
9.99 998 29
II
II
0
log cot
[
1
742 1.77280
730 1.76538
9.99994
9.99994
9.99993
1
1
I cd I
1.75808
log tan
I
log s:n
" I'
60
59
58
57
56
61 .
9.99 998 28
1297 2.01 775 9.99 998 27
125912.00478
9.99998 26
5.31 441,,800781 ,i 11H 1.
5.31 441 " 8.02 004
1190' 1.97 996 9.99 998 24
5. 31 441 8.03 194 1159
1.96 806 9.99 997 23
5.314418.0435311281.956479.9999722
5.31 441 8.05 481 1100 I. 94 519 9.99 997 21
5.31 441 8.06 581 1072 1.93 419 9.99 997 20
5.314408.0765310471.923479.9999719
5.31 440 8.08 700 1022 I. 91 300 9.99 997 ] 8
5.31 440 8.09 722
998 1.90 278 9.99 997 17
5.31440
8.10720
9.9999616
976 1.89280
5.31 440 8.11 696
955 1.88 304 9.99 996 15
5.31 440 8. 12 651
934 1.87 342 9.99 996 14
5.31 440 8. 13 585
915 1.86 415 9.99 996 13
5.31 440 8. 14 500
895 1.85 500 9.99 996 12
5.31 440 8. 15 395
878 I. 84 60; 9.99 996 11
5.31439
8.16273
9.99995 10
860 1.83727
5.31 439 8. 17 133
843 I. 82 867 9.99 995 9
5.314398.17976
8281.820249.99992
8
5.31439
8.18804
7
812 1.81 196 9.99995
5.31 439 8. 19 616
797 I. 80 384 9.99 995 6
5. 31 439 8. 20 413
782 1.79 58Z 9.99 994 5
5.31439
8.21 195
76911.78805
9.99994
4
5.31439
8.21964
756 1.78036 9.99994 3
5. 31 441 7.98 225
5.31 441117.99522
976 5.31 444
954 5.31 444
93415.31 444
914 5.31 444
896 5. 31 444
877 5.31 444
860 5. 31 444
843 5.31 444
827 5.31444
812 5.31 444
797 5.31 444
782 5.31 444
769 5.31441
755 5.31442
743 5.31 442 5.31438' 8.22 720
730 5.31 445 5.3143818.23462
5.31445
5.31 4381 8.24192
1
r
TABLE III
10;: tan
2
1
0
I'
.
r~
.1
1
,i
lug sin
cpl S
d
cpl T
II
log tan
I cd i
91° 18102710
log cot
I
lo~
3600 0 8.24186
5.31 44~ 5.314388.241927181.758089.9999360
3660 1 8.24 903 717 5.31 442 5.314388.249107061.750909.9999359
706
3720 2 8.25609
5. 31 438 8.25 616
695 5.31442
696 I. 74 384 9.99 993 58
3780 3 8.26304
5.31445
5.31438
8.26312
9.99993 57
684
684 1.73688
3840 4 8.26988
,5.31
445
5.314378.269966731.730049.9999256
673
3900 5 8.27 661
5.31
445
5.314378.276696631.723319.9999255
3960 6 8.28 324 663 5.31 445 5.314378.283326541.716689.9999254
4020 7 8.28 977 653 5.31 445 5.31437
8.28986
9.99992 53
644
643 1.71014
4080 8 8.29621
5.31437
8.29629
1.70371
9.99992 52
634 5.31445
4140 9 8.30255
5.31437
8.30 263
9.99991 51
624 15.31445
~~~ 1.69737
420010 8.30879
5.31446
5 . 31 437 8. 30 88§
I. 69 112 9. 99 991 50
616
617
4260 II 8.31 495
5.31 446 5.31436
8.31505
1.68495
9.99991 49
4320 12 8.32 103 608 5.31 446 5. 31 436 8.32 112 607 I .67 888 9.99 990 48
599
599
4380 13 8.32 702
5.31
446
5.314368.32711591
1.672899.9999047
4440 14 8.33 292 590 5.31 446 5.31 436 8.33 302
583
584 I. 66 698 9.99 990 46
450015 8.33875
'5.31446
5.31436
8.33886
9.9999045
575
575 1.66114
4560 16 8.34 450 568, 5.31 446 5.31435
8.34461
1.65539
9.99989 44
568
4620 17 8.35 018
,5.31
446
5.31435
8.35 029
9.99989 43
561 1.64971
4680 18 8.35 578 560 I 5.31 446 5.314328.355905531.64410
9.9998942
4740 19 8. 36 131 553 i 5. 31 446 5.31 43~ 8.36143
9.99989 41
547
546 1.63857
4800 0 8.36 678
5.31 446 5.31 435 8.36 689
539
540 I. 63 311 9.99 988 40
486021 8.37217
5.31434
8.37229
1.62771
9.99988 39
533 5.31447
533
492022 8.37750
5.31434
8.37762 527 1.62238
9.99988 38
526 15.31447
498023 8.38276
.5.31447
5.31434
8.38289
1.61711
9.9998737
504024 8.38 796 520 5.3 I 447 5.31 434 8.38 809 520 I. 61 191 9.99 987 36
514
514
5100 5 8.39310
5.31447
5.31 434 8.39 323
I. 60 677 9.99 987 35
516026 8.39 Pl8 508 5.31447
5.31 433 8.39 832 509 I. 60 168 9.99 986 34
502
502
522027 8.40 320
5.31
447
5.31433
8.40334
496
496 1.596669.9998633
528028 8.40816
5.31447
5.31 433 8.40830
9.99986 32
491
491 1.59170
534029 8.41307
5.31447
5.31433
8.41321
9.99985 31
485
486 1.58679
5400 0 8.41792
5.31433
8.41807
1.58193
9.99985 30
480 5.31447
480
54603\ 8.42272
5.31448
5.31432
8.42287
9.9998;
29
474
475 1.57713
552032 8.42746
5.31448
5.314328.427624701.572389.9998428
558033 8.432161464470 115.31448 5.31432
8.43232
1.56768
9.99984 27
564034 8.43680 1459 115.31448
5.31432
8.436961 '464
A;;'; 1 56304 9 99984 26
j
I
.44139 455 5,31448 5 31 431 8.44156 I
9.99983 25
455 '1.55844
576036 8.44594
5.31448
5.31 431 8.44 611
1.55 389 9.99 983 24
582037 8.45 044 450 5.31 448 5.31 431 8.45 061 450 1.54 939 9.99 983 23
5880 38 8.45 489 445 5.31 448 5.31 431 8.45 507 446 1 .54 493 9.99 982 22
594039 8.45 930 441 5.31 449 5.31 431 8.45948 441 1.54052
9.99982 2]
436
437
6000
8.46366
5.31 449 5.31430
8.4638;
9.99982 20
432 1.53615
606041 8.46 799 433 5.31 449 5.31430
8.46817
427
428 1.53 18~ 9.99981 19
612042 8.47226
8.47245 424
1.52755
9.99981 18
424 5.3] 449 5.31430
618043 8.47650
5.31449
5.31430
8.47669
9.99981 17
420 1.52331
6240 44 8. 48 069 419 5 . 31 449 5.31 429 8.48089
1.51 911 9.99980 16
416
416
6300 5 8.48 485
449 5.31429 8.48505 412 1.5149; 9.99980 15
6360 46 8.48 896 411 I 5.31
5.31 419 5.31429
8.48917
408
408 1.51 08~ 9.99979 14
642047 8.49304
5.31428
8.49325
9.99979 13
5.31420
404 1.50675
648048 8.49 708 404
8.49729 401 1.50271
9.99979 12
5.31 450 5.31428
6540 49 8.50 108 400 5.31
450
5.31
428
8.50
130
396
397 I. 49 870 9.99 978 11
6600
8 . 50 504
5 . 31 450 5.31 428 8.50 527 393 1. 49 473 9.99 978 10
6660 51 8.50 897 393
5.31 450 5.31 427 8.50 920 390 1. 49 080 9.99 977 9
6720 52 8.51 287 390
8.51310
5.31 450 5.31427
386 1.48690
9.99977
8
678053 8.51 673 386
382
5.31 450 5.31 427 8.51 696 383 I .48 304 9.99 977 7
684054 8.52055
5.31427
8.52079
379 5.31450
380 1.47921
9.99976
6
6900 5 8.52434
5.31426
8.52452
376 5.31451
376 1.47541
9.99976
5
696056 8.52810
5.31426
8.52835
373
5.31451
373 1.47165
9.99972
4
702057 8.53 183 369
8.53208
5.31 451 5.31426
370 1.46792
9.99975
3
708058 8.53 552 367
5.31425
8.535/§
5.31 451
367 1.46422
9.99974
2
714059 8.53919
5.31425
8.53945
363 5.31451
363 1.46 055 9.99974
I
7200
8.54 282
5.31 451 5.31425
8.54308
1.45692
9.99974
0
logcos
d II
log cot I c d I log tan J log sin
() I
1
.
I
I
67
~
20
sin
cpl
d
7200 0 8.54282
726 1 8.54 642
7320 2 8.54 999
7380 3 8.55 354
7440 4 8.55705
7500 5 8.56 054
7560 6 8.56400
762 7 8.56743
768 oS 8.57 084
774 9 8.57421
780010 8.57757
786 II 8.58089
792 12 8.58419
798 13 8.58747
804 14 8.59 07~
810 15 8.59 395
816 16 8.59 715
822 17 8.60 033
828 18 8.60 349
834 19 8.60662
840 20 8.60973
846 21 8.61282
852022 8.61589
858023 8.61 894
864 24 8.62196
870025 8.62 497
876 26 8.62795
882 27 8.63 091
888 28 8.63 385
894 29 8.63678 i
900 30 8.63968
906 31 8.64 256
912 32 8.64543
918 33 8.64827
9240 34 8.65 110 i
S
cpl T
II
log tan
5.31451 5 . 31 4 2 ~ 8. 54 308
5.31 451 5.31425
8.54669
5.31 452 5.31 424 8.55 027
8.55382
5.31 452 5.31424
360
357
355
351
5.31 452
5.31452
5.31.452
5.31 453
5.31453
5.31453
5.31453
5.31453
5.31453
5.31454
5.31 454
5.31 454
I5.31 454
5.31 454
5.31 45~
5.31 45~
5.31 45~
5.31455
5.31 455
5.31455
5.31 455
5.31456
5.31 456
5.31 456
5.31 456
5.31456
5.31 456
5.31457
5.31457
I 5.31 457
11
II
1
I log
cas
I
/
349 5.31 452
5.31 424 8.55 734
5.31423
8.56083
346
5.31423
8.56429
343
5.31423
8.56773
341
5.31422
8.57114
337
5.31422
8.57452'
336
5.31422
8.57788
332
5.31 421 8.58 121
330
5.31421
8.58451
328
5.31421
8.58779
325
5.31 421 8.59 105
323
5.31420
8.59428
320
5.31420
8.59749
318
5.31 420 8.60 068
316
5 . 31 419 8. 60 384
313
5.31 419 8. 60 698
311
5.31 418 8.61 009
309
5.31 418 8.61 319
307
5.31418
8.61626
305
5.31 417 8. 61 931
302
5.31417
8.62234
301
5.31 417 8.62 535
298
5.31 416 8.62 834
296
5.31 416 8.63 131
294
5.31416
8.63426
293
5.31 415 8.63 718
290
5.31412
8.64009
288
5.31 415 8.64 298
287
5.31 414 8. ~4 585
284
5.31 414 8,4 870
283
5.31 413 8.65 154
281
8.6-' 391 I 279 II S. 31 457 5.31413 II 8.0.1 43~:
9360 36 8.65 b70 277'. 5.31 457 5.314131 8.65715
942037 8.65947
8.65993
276 5.31458 5.31412
948 38 8.66223 274 5.31458 5.31412
8.66269
954 39 8.66497 272 5.31458 5.31412
8.66543
960040 8.66769 270 5.31458 5.31411
8.66816
9660 41 8.67 039 269 5.31 458 5.31 411 8.67 087
972 42 8.67 308 267 5.31 459 5.31 410 8.67 356
978043 8.67575
984 44 8.67 841
990 45 8.68104
996 46 8.68 367
1002 47 8.68627
1008 48 8. 68 886
1014 49 8. 69 144
1020050 8.69 400
1026 51 8.69 654
1032 52 8.69 907
1038 53 8.70 159
1044 54 8. 70 409
1050 55 8. 70 65~
1056 56 8.70 905
I 062 57 8. 71 151
1068 58 8. 71 395
1074 598.71638
I080 60 8. 71 880
I cd
266
263
263
260
259
258
256
254
253
252
5.31459
5.31 459
5.31459
5.31 459
5.31460
5. 31 460
5. 31 460
5.31 460
5.31 460
5.31 461
249
247
246
5 . 31 461
5.31 461
5.31 461
243
242
5.31 462
5.31462
5. 31 462
250 5.31 461
244 5.31 462
d
II
5.31410
5.31 410
5.31 409
5.31 409
5.31 408
5.31 408
5.31 408
5.31407
5.31 407
5.31 406
5.31406
5.31 405
5. 31 402
5.31 405
5.31404
5.31404
5.31403
5.31 403
8.67624
I
8.67 890
8. 68 154
8.68 417
8.68 678
8. 68 938
8.69 196
8.69453
8. 69 708
8.69 962
8.70211
8.70 465
8. 70 714
8.70 962
8.71208
8.71453
8.71697
8.71 940
I
log cot
I
I
TABLE III
I .32 110
1.31 846
I .31 583
1.31 322
I. 31 062
I .30 804
1.30547
I .30 292
1.30 038
1.29786
I .29 535
1.29 286
1.29 038
1.28792
1.28547
1.28303
1.28060
log tan
3"
d
log sin
266 1.32376
cd
.
log cot
361 I. 45 692
358 1.45331
355 I .44 973
352 1.44618
349 1.44 266
346 1.43917
344 1.43571
341 1.43227
338 1.42886
336 1.42548
333 1.42212
330 1.41 879
328 1.41549
326 1.41221
323 I .40 895
321 1.40572
319 1.40251
316 1.39 932
314 I . 39 616
311 1.39 302
310 1.38 991
307 I .38 681
305 1.38374
303 I. 38 069
301 1.37766
299 1.37 465
297 1.37 166
295 1.36 869
292 1.36574
291 1.36 282
289 1.35991
287 1.35 702
285 I .35 415
284 1.35 130
281 I 1.34 846
280 I 1.34 )tJ)
278. 1.34285
276 1.34007
274 1.33731
273 1.33457
271 1.33184
269 I. 32 913
268 I . 32 644
264
263
261
260
258
257
255
254
252
251
249
248
246
245
244
243
1820 272°
log tan
cd
I
log cot
0 8.718802408.7194012411.28060
1 8.72 120
239 8.72 181 239 1.27819
2 8.72359
238 8.72420 239 I. 27580
3 8.72597 237
8.72659
237 1.27341
4 8.72834
235 8.72896 236 1.27 104
5 8.73069 234 8.73 132
234 1.26868
6 8.73303
232 8.73 366 234 I. 26 634
7 8.73535 232 8.73600
232 1.26400
8 8. 73 767
230 8. 73 832 231 1.26 168
9 8.73 997 229 8.74 063
229 1.25 937
10 8.74226
8.74292
228
229 1.25708
11 8.744542268.745212271.25479
12 8.74680 226 8.74748
1.25252
13 8.74 906 224 8. 74 974 226 1.25 026
225
14 8. 75 130 223 8.75 199
224 1.24 801
15 8.75 35~ 222 8.75423
1.2457Z
222
16 8.75575 220 8.75645
1.24355
222
17 8.75795 220 8.75867 220 1.24133
18 8.76015 219 8.76087 219 1.23913
19 8.76234 217 8.76 30~ 219 1.23694
208.764512168.765252171.23475
21 8.76667 216 8.76742 216 1.23258
22 8.76883 2148.76958
215 1.23042
23 8.77 097 213 8.77 173 214 1.22827
24 8.77310 212 8.773871213
1.22613
25 8.77522 211 8.77 600 211 1.22400
26 8.77 733 210 8. 77 811 211 1.22 189
27 8.77943 209 8.78022 210 1.21978
28 8.78152 208 8.78232 209 1.21 768
29 8.78 360 208 8.78 441 208 1.21 559
30 8.78 568 206 8.78 649 206 1.21 351
31 8.78774 205 8.78855 206 1.21 145
32 8.78 979 204 8.79 061 2051 1.20 939
33 8.791831203
8.79266 20411.20734
.
.
7
1
9.99 955
9.99955
9.99 954
9.9995421
9.99953
9.99952
9.99 952
9.99 951
9.9995116
9.99950
9. 99 949
9. 99 949
9. 99 948
24
23
22
9 . 99 948
11
39
19
18
17
5
14
13
12
9.99947 10
9. 99 946 9
9. 99 946 8
9. 99 945 7
9. 99 944 6
9. 99 944 5
9. 99 943 4
9.99 942 3
9.99942
2
9. 99 941 1
9. 99 940 0
log sin
'35
36
37
38
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
8.795881;01
930 1830 2780
Prop. Parts
log cas
9. 99 940 60
9.99 940 59
9.99 939 58
9.99938 57
9.99 938 56
9.99937 55
9.99 936 54
9. 99 936 53
9.99 935 52
9.99 934 51
9. 99 934 60
9.99 933 49
9.99 932 48
9.99932 47
9. 99 931 46
9.99 930 45
9.99 929 44
9.99929 43
9.99928 42
9.99 927 41
9.99926 40
9.99 926 39
9.99 925 38
9.99924 37
9.99 923 36
9.99923 35
9.99 922 34
9.99 921 33
9.99 920 32
9.99 920 31
9.99 919 30
9.9991829
9.99 917 28
9.99 917 27
8.79673120211.20 32Z 9.99915
8.79789 201 8.79875 201 1.20125
8.79990 199 8.80076 201 1.19924
8.80 189 199 8.80 277 199 1. 19 723
8.80388 197 8.80476 198 1.19524
8.80 585 197 8.80 674 198 1. 19 326
8.80 782 196 8.80 872 196 I. 19 128
8.80978 195 8.81068
196 1.18932
8.81 173 194 8.81264
195 1.18736
8.81 367 193 8.81 459 194 1. 18 541
8.81560 192 8.81653
193 1.18347
8.81 752 192 8.81846
192 1.18154
8. SI 944 190 8.82 038 192 1. 17 962
8.82 134 190 8.82 230 190 1. 17 770
8.82 324 189 8.82 420 190 1. 17 580
8.82 513 188 8.82 610 189 1. 17 390
8.82701 187 8.82799
188 1.17201
8. 82 88~ 187 8.82 98Z 188 I. 17 013
8.83075 186 8.83175
186 1.16825
8.83 261 185 8.83 361 186 1. 16 639
8.83446 184 8.83547
185 1.16453
8.83 630 183 8.83 732 184 I. 16 268
8.83813 183 8.83916
184 1.16084
8.83 996 181 8.84 100 182 1. 15 900
8.84177 181 8.84282
182 1.15718
8.84 358
8.84 464
1. 15 536
1
log
920
1
TABLE III
9.99914
9.99913
9. 99 913
9. 99 912
9.9991120
9.99910
9.99909
9.99 909
9. 99 908
9.99 907
9. 99 906
9.99 905
9. 99 904
9 . 99 904
26
24
23
22
21
19
18
17
I6
16
14
13
12
11
9.99 903 10
9.99 902 9
9.99 901 8
9. 99 900 7
9. 99 899 6
9. 99 898 5
9.99 898 4
9.99 897 3
9.99 896 2
9.99895
1
9. 99 894 0
2U
239
231
235
23i
I
4.0
4.0
4.0
3.9
3.9
2
8.0
8.0
7.[1
7.8
7.8
3
12.0 12.0 11.8 11.8 11.7
4
16.1 15.9 15.8 15.7 15.6
5
20.1 1~.9 19.8 19.6 19.5
6
24.1 23.9 23.7 23.5 23.4
7
28.1 27.9 27.6 27.4 27.3
8
32.1 31.~ 31.6 31.3 31.2
9
36.2 35.8 35.6 35.2 35.1
10
40.2 39.8 3~.5 39.2 39.0
20
80.3 79.7 79.0 78.3 n.o
30 120.5 11~.5 11S.5 117.5 117.0
40 160.7 1.';9.:! 1.';8.0 156.7 156.0
50 200.S 199.2 197.5 195.8 195.0
232
229
221
225
223
1
3.9
3.8
3.8
3.8
3.7
2
7.7
7.6
7.6
7.5
7.4
3
11.6 11.4 11.4 11.2 11.2
4
15.5 15.3 15.1 15.0 14.9
5
19.3 19.1 18.9 18.8 18.6
6
23.2 22.9 22.7 22.5 22.3
7
27.1 26.7 26.5 26.2 26.0
8
30.9 30.5 30.3 30.0 29.7
9
34.8 34.4 34.0 33.8 33.4
10
38.7 38.2 37.8 37.5 37.2
20
77.3 76.3 75.7 75.0 74.3
30 116.0 114.5 113.5 112.5 111.5
40 154.7 152.7 151.3 150.0 148.7
50 193.3 190.8 189.2 187.5 185.8
222
220
217
215
213
1
3.7
3.7
3.6
3.6
3.6
2
7.4
7.3
7.2
7.2
7.1
3
11.1 11.0 10.8 10.8 10.6
4
14.8 14.7 14.5 14.3 14.2
5
18.5 18.3 18.1 17.9 17.8
6
22.2 22.0 21.7 215
21.3
7
25 9 25.7 25.3 25.1 24.8
8
29.6 29.3 28.9 28.7 28.4
9
33.3 33.0 32.6 32.2 32.0
10
37.0 36.7 36.2 35.8 35.5
20
74.0 73.3 72.3 71.7 71.0
30 111.0 110.0 108.5 107.5 106.5
40 148.0 146.7 144.7 143.3 142.0
50 185.0 183.3 180.8 179.2 177.5
211
208
206
203
201
1
L5
L3
3.4
L4
L4
2
7.0
6.9
6.9
6.8
6.7
3
10.6 10.4 10.3 10.2 10.0
4
14.1
13.9
13.7
..,
...
... <) 13.5 13.4
.
6
21.1 20.8 20.6 20.3 20.1
7
24.6 24.3 24.0 23.7 23.4
8
28.1 27.7 27.3 27.1 26.8
9
31.6 31.2 30.9 30.4 30.2
10
35.2 34.7 34.3 33.8 33.5
20
70.3 69.3 68.7 67.7 67.0
30 105.5 104.0 103.0 101.5 100.5
40 140.7 138.7 137.3 135.3 134.0
50 175.8173.3171.7169.2167.5
199
191
195
193
192
1
3.3
3.3
3.2
3.2
3.2
2
6.6
6.6
6.5
6.4
6.4
3
10.0
9.8
9.8
9.6
9.6
4
13.3 13.1 13.0 12.9 12.8
5
16.6 16.4 16.2 16.1 160
6
199
19.7 19.5 19.3 19.2
7
23.2 23.0 22.8 22.5 22.4
8
26.5 26.3 26.0 25.7 25.6
9
29.8 29.6 29.2 29.0 28.8
10
33.2 32.8 32 5 32.2 32.0
20
66.3 65.7 65.0 64.3 64.0
30
99.5 98.5 97.5 96.5 96.0
40 132.7 131.3 130.0 128.7 128.0
50 165.8 164.2 162.5 160.8 160.0
189
187
185
183
181
1
3.2
3.1
3.1
3.0
3.0
2
6.3
6.2
6.2
6.1
6.0
3
9.4
9.1.
9.2
9.2
9.0
4
12.6 12.5 12.3 12.2 12.1
5
15.8 15.6 15.4 15.2 15.1
6
18.9 18.7 18.5 18.3 18.1
7
22.0 21.8 21.6 21.4 21.1
8
2.'\.2 24.9 24.7 24.4 24.1
9
28.4 28.0 27.8 27.4 27.2
10
31.5 31.2 30.8 30.5 30.2
20
63.0 62.3 61.7 61.0 60.3
30
94.5 93.5 92.5 !H.5 90.5
40 126.0 124.7 123.3 122.0 120.7
50 157.5 155.8 154.2 152.5 150.8
ProD. Parts
Q8
:1.760 266'
3560
86Q
69
l
40
TABLE III
r
log sin
d
log tan
log cot
cd
log cos
'0 8.84 358
8.84
464/182 1. 15 536
1 8.84 539 181
179 8.84 646 180 1. 15 354
2 8.84718
8.84826
3
1.15174
180
1.14994
179
1788.850021179
8.84897
940
9.99892
9.9989157
182
1
-!
11
12 8.86474
13 8 . 86645
14 8.86816
171 8.865911 172 1.13409
8 86763
171'
172. I 13237
171 8.86935 171 1.13065
9.9988348
9 . 99882 47
9.99881 46
3
4
5
6
7
15 8.86 987 169 8.87 106 171 1. 12 894 9.99 880 45
16 8.87 156
8.87 277
1
1170 1. 12 723
169
169 8.87447
17
8.87325
18
8.874941678.876121691.123849.99878
16911.12553
9.99 879 44
9.99879 43
10
42
19 8.876611688.87785
168 1.12215
20 8.87829 166 8.87953 167 1.12047
21 8 . 87995 166. 8 88 120
167' 1 11 880
22 8.88161 165 8.88287 166
1.11713
23 8.88326 164 8.88453
165 1.11547
8.88654
8.88817
27
8.88 980
1638.88783
163 8.88948
162 8.89
165 1.11217
163 1.11052
30 8.89 464 1 8 89 598
31 8.89625 - 6 I.
159 8.89760
784
161
8.89
33
8.899431598.900801601.09920
34
35
36
8.90102
8.90260
8.90417
37
8.90574
159 8.89
920
160
1.10563
9.99867
31
1 10402 9.99 866 30
9.99865
I. 10 080
1.09285
11.9
2
11.8
~§j:~§U §U §g
3
~g,g.
8.91040
155 8.91029
8.91 195
8.91349
8.91 502
8.91655
155 891
156 1.08971
185 155 1.08815
155 1.08505
153 1.08350
154 1.08197
J53> 1 . 08 043
15211.07890
48
49
50
51
8.92 261 150 8.92414 151 1.07586
8.924111508.925651511.074359.99846
8.92 561 149 8.92 716 150 1.07 284
8.92710 1498.92866
150,1.07134
52
8.92859
1488.93016
57
8.93594
1.06984
170"
26j" 3650
log tan
28:8
28:7
.
.
..
14.2
14.1
14.0
2.8
5.5
18/;
2.5
5.5
164
2.~\
5.:'
13.9
183
182
2.7
5.4
In
16
9.99857
21
9.99856
20
1(j.(j
§j:?
55.3
](;.5
lU,4
2.7
5.4
1(j.3
16.2-.
§j:~ §i:~ §U
55.0
54.7
~g 1~3:g ]~~:g ]~~:~ ]~u
.
".'
<:
~-
~j:3
54.3
19
20
21
22
23
24
25
26
8. 96 68~ 136
8.96825/135
8.96 96Q 135
8.97095 134
8.97 229 134
8.97 363 133
8.97496 133
8.97629 133
27
28
29
30
31
32
37
38
39
54.0
1~~:8
12 2'1 53.7 53.;'0 53.Q 52.7 52.3
11 ~8 1~~:L~2S16g:81b~:~16H
50 134.2 133.3 132.5 131.7 130.8
9.99 845 10
9.99844
9
1 12~~ 1:'~ Vt v~ In
48
49
50
51
9.00456
9.00581
9.00 704
9.00 828
52
9.00 951
15
14
9.99842
8
2
7
~16'~16'~ 16'~ 16.g 16'~
513:012:912:812:812:7
~[)"
sin
5.1
5.1
~}~:g }~'f }~:6 g:~
}~:?
20.7
20.5
20.4
20.3
51.7
51.3
51.0
50.7
16
2
20
~8
52.0
'
5.1
20.8
3
I
55
56
57
58
59
60
1bt816~L6~'L6g:L6~S
70
8.99450
8.99577
8.99704
8.99830
8.99956
9.00 082
9.00 207
9.00332
9.01
§~.g §U
Prop. Parts
8.96877
8.97013
8.97 150
8.97285
8.97421
8.97556
8.97691
8.97825
8.97959
8.98092
8.98 225
8.98 358
8.98 490
8.98622
8.98753
8. 98 88~
..
I
1.05805
145
1.05 660
145
1.05515
145
1.05 370
143
1. 05 227
9.99 834 60
9.99 833 59
9.99 829 55
9. 99 828 54
144
9.99 832 58
9.99 831 57
9.99 830 56
142 1. 04 798
142 1 .04 656
141 1.04514
140 1.04373
141 1.04233
139 1 . 04 092
140 1.03 953
138 1.03813
9.99827
9.99 825
9. 99 824
9.99 823
9.99 822
9. 99 821
9.99 820.
9. 99 819
9.99817
9.99816
9. 99 815
9.99814
53
52
51
50
49
48
47
46
45
44
43
42
1.03 123
1.02 987
1.02 850
1.02715
1.02579
I .02 444
1.02 309
1.02 175
9. 99 813
9.99812
9.99810
9. 99 809
9. 99 808
9.99 807
9. 99 806
9. 99 804
41
40
39
38
37
36
35
34
9.99 803
9.99 802
9.99801
9. 99 800
9. 99 798
9.99 797
9. 99 796
~9.99795
9.99793
9. 99 792
33
32
31
30
29
28
27
26
25
24
139 1. 03 675
138 1.03536
137 I. 03 398
138 1.03261
136
137
135
136
135
135
134
134
133 1.02041
133 1.01 908
133 1.01 775
132 1. 01 642
132 1.01 510
13111.01378
131 1.01247
11311
1. 0 I 11
8.99015130'
1.0098~
8.99145
130 1.00 85~
127 8.99662
127 8.99791
126 8.99 919
126 9.00046
126 9.00174
125 9.00301
125 9.00427
124 9.00553
129 1.00338
128 1. 00 209
127 1.00081
128 0.99 954
127 0.99826
126 O. 99 699
126 0.99573
126 0.99447
9.99 787 20
9.99 786 19
9.99 785 18
9.99 783 17
9.99 782 16
9.99781
16
9 . 99 780 14
9.99778
13
123 9.01
124 O. 98 945
0.98821
9.99773
9.99 772
9
8
9.99771
7
9. 99 769
6
125 9.00 679
123 9.00805
124 9.00930
123 9.01055
179
9.99 777 12
126 0.99321
125 O.99 19~ 9.99776 11
125 0.9907& 9. 99 775 10
53 9.01074 122 9.01303 124
124 0.98697
54
50 130.0 129 2 128.3 127.5 126.7
0
I
5.2
§~:6 §~'§ §g
1
9.99834
log
8
5.2
8.95 202
8.95 344
8.95 486
8.95627
8.95767
8.95908
8.96047
8.96187
8.96325
8.96464
8.96 602
8.96739
I
log cos
8.99066 128 8. 99 27~ 130 1. 00 725 9.99791 23
8.99 194 128 8.99405 129 1.00595
9. 99 790 22
8.99 322 128 8. 99 534 128 I. 00 466 9.99 788 21
40
41
42
43
44
45
46
47
18
17
16
141
140
140
139
139
139
138
138
137
137
136
136
8.97762
132
8.97894
132
8.98 026 131
8.98157
131
8.98 288 131
8.98419
130
33 8.98 549 1130
8
986791129
~'-4
8 98 808 ; 129
'35
36 8.98937
129
50 138.3137.5136.7135.8135.0
1
1:'~ 12~~ 1:'~ 12~~ Il~
2
5.4
5.3
5.3
5.3
5.2
~l~:g 1~:g l~:g 16:~ 16:~
5
13.4 13.3 13.2 13.2 13.1
6
16.1
16.0
15.9
15.8
15.7
7
18.8 18.7 18.6 18.4 18.3
8
21.5 21.3 21.2 21.1 20.9
§U §;§:g ~gj §~:~ §~:g
9.99843
I
18
17
~~'3
23
9.99836
1.05805
29:0
~~.~ ~~'8 ~n
24
9.99838
i
29:2
~~'6
9.99859
146 (05951
led
29:3
1!J.2 IV. 1 ID.l! 18.9
22.0 21.9 21.7 21.6
147 1:06244
log cot
~§~.~ §~.~ §~'I §~'b §~'g
1
7,19.1
8
22.1
145 8.94049
8.94195
14.3
1t6
1~:~ 131\
1~:~ lU
13.~ lU
13.1;
146 8.93756
Id
14.4
r, ~I 13.~
58 8.93740 145 8.93903 146 1 06097 9.99837
log cos
H.5
17.4 17.3 17.2
20.3 20.2 20.1
8
147 8.93462
8.94030
8.~
11.
26
1.06538
9.99840
5
56 8.93448 146 8.93609 147
147 1 06391 9.99839 4
8.93885
14.6
8.6
11.5
143
8.95029
8.95 170
8.95 3~0
8.95 450
8.95 589
8.95 728
8.95 867
8.96 005
8.96143
8.96280
8. 96 417
8.96553
7
8
9
10
11
12
13
14
15
16
9.99847
548.931541478.933131491.066879.998416
8.93301
14.7
17.6 17.5
20.,} 20.4
8.7
11.6
9.99862
9.9986125
9.99860
9.99854
9.99853
9.99852
9 . 99 851
9.99850
149
53 8.93007 147 8.93165 148
1.06835
55
8.8
11.7
27
154 8: 91 340 155 1.0866Q 9.99855 19
153 8.91495
153 8.91 650
152 8.91803
8 . 91 807 152. 8 91 957
8.91959
151 8.92 110
8.8
11.7
9.99863
478.921101518.9226215211.077389.998481316
60
8. 94 746 141 8.94 917 143 1.05 083
8.94887 142 8.95060
142 1. 04 940
11.9
~M:b g:g }g3 }~:~ }H
22.8 2g.7 22.5 22.4 22.3
20
log cot
5
6
12.1
14.2
(;
cd
21.2 21.1 20.9 20.8 20:6
17g
3
5
21[
log tan
4
12.1
166
29
9.99 864 28
157
38 8.90730 156
155 8.90872 157 1.09128 9.99858 22
39 8.90885
59
3.0
16 §~:~ §gj §~:~ §~:6 §~:g
20
57.0 56.7 56.3 56.0 55.7
~8 1n:g1~~:~1~~:~1~~:81~U
50 142.5 141.7 140.8 140.0 139.2
9.99868 32
158 8.90240115911.09760
1578.90399
15811.09601
157 8.90557
158 1.09443
8.90715
35
34
9.99 869 33
162. 1.10240
160
32
46
3.0
~:8 ~:8 ~:8 fx
9.9987741
4011i3116:7116:011.';:3114:7
9.99876 40 50 146.7145.8145.0144.2143.3
2.8 170
2.8 169
2.8 168
2.8 187
2.8
9 . 99875 39 1 171
9.99874 38 2 5.7 5.7 5.6 5.6 5.6
9.99873 37
~1~:~ &~ 1~:§ 1t~ IN
9.99871
9.99870
111 163 1. 10 889
28 8.89142 162 8.89274 163 1.10726
29 8.89304
1608.89437
45
3.0
~J
0
I
950 1850 275-
5°
d
~}~:§}~:} }j:§ }U }H
3.0
~8
24 8.88 490 164 8.88 618 165 1. 11 382 9.99 872 36
25
26
42
43
44
178
1758.85717176
1.14283
9.9988853
10 30.330.229.8
29.7 29:5
8.85 78Q
8.85 893
J. 14 107 9.99 887 52
~8
~?:6 gR~
~8:~ ~8:3 ~g:g ';j
175
176
8.85955
1738.86069
1.13931
9.9988651
401'?1.3120.711'1.3118.7118.0
174
8. 86 1281 8 86 243
I 13757 9 99885 50 50 1,,1.7 150.8 149.2 148.3 147.5
1173'
174"
178 175 174 173 171
8.86301
173 8.8641717411.13583
9.99884
49
1
2.9
2.9 5.8
2.9 2.9
2.9
2
5.9
5.8
5.8
5.7
10
4]
7
179
log sin
8.94 030
8.94195
144
8.94174
8. 94 340
143
8.94 317
8.94 485
144
8.94630
8.94461142
8.94603
8.94773
5
58
181
8.85605
8
9
40
TABLE III
I
Pro!'. Parts
9.99 894 60
9.99 893 59
4 8.85075 177 8.85185 178 1.14815 9.99891 56
5 8.85252 177 8.85363
1.14637
9.99890 55
6 8.8542~ 1768.85540:177177 1.144609.99889
54
7
1840 2'140
196 122 9.01427
123 0.98573
9.01 318 122 9.01 550 123 0 . 98 450 9.99768
9.01 440 121 9.01 673 123 0.98 327 9.99767
5
4
9.01 682 121 9.01 918 122 0.98082
9.01 803 120 9.02040 122 0.97 960
9.01923
9.02162
0.97838
log cos I d
log cot
log tan
Icdl
9.99765
3
9.99764
9.99763
9.99761
2
1
0
9.01
561
121 9. 01 796
1740 2640 354u
122 O. 98 204
Prop. Parts
1
2
3
4
5
6
7
8
9
10
20
30
40
50
1
140 139
2.3 2.3
138 137 138
2.3 2.3 2.3
4
9.3
9.2
9.1
6.2
6.2
6.1
Prop.
Parts
I
2
3
4
5
6
7
8
9
10
20
30
40
50
2
3
5
6
4.7
7.0
4.6
7.0
9.3
145
2.5
4.9
7.4
9.9
12.3
14.8
17.3
19.7
22.2
24.7
49.3
74.0
98.7
123.3
143
2.4
4.8
7.2
9.5
11.9
14.3
16.7
19.1
21.4
23.8
47.7
71.5
95.3
119.2
4.6
6.9
147
24
4.9
7.4
9.8
12.2
14.7
17.2
19.6
22.0
24.5
49.0
73.5
98.0
122.5
142
2.4
4.7
7.1
9.5
11.8
14.2
16.6
18.9
21.3
23.7
47.3
71.0
94.7
118.3
148
2.4
4.9
7.3
9.7
12.2
14.6
17.Q
19.5
21.9
24.3
48.7
73.0
97.3
121.7
141
2.4
4.7
7.0
9.4
11.8
14.1
16.4
18.8
21.2
23.5
47.0
70.5
94~
117.5
151 149
2.5
2.5
5.0
5.0
7.6
7.4
10.1 9.9
12.6 12.4
15.1 14.9
17.6 17.4
20.1 19.9
22.6 22.4
25.2 24.8
50.3 49.7
71>.5 74.5
100.7 99.3
125.8 124.2
145 144
2.4
2.4
4.8
4.8
7.2
7.2
9.7
9.6
12.1 12.0
14.5 14.4
16.9 16.8
19.3 19.2
21.8 21.6
24.2 24.0
48.3 48.0
72.5 72.0
9~7 9~0
120.8 120.0
4.6
6.8
4.5
6.8
9.1
11.7 11.6 11.5 11.4 11.3
14.0 13.9 13.8 13.7 13.6
7 16.3 16.2 16.1 16.0 15.9
8 18.7 18.5 18.4 18.3 18.1
9
21.0 20.8 20.7 20.6 20.4
10
23.3 23.2 23.0 22.8 22.7
20
46.7 46.3 46.0 45.7 45.3
30
70.0 69.5 69.0 68.5 68.0
40
93.3 92.7 92.0 91.3 90.7
50 116.7 115.8115.0 114.2 113.3
135 134 133 132 131
2.2
2.2
2.2
2.2
2.2
4.5
4.5
4.4
4.4
4.4
~I 6.8
6.7
6.6
6.6
6.6
41 9.0
8.9
8.9
8.8
8.7
.'i 11.2 11.2 111 11.0 ]0.\1
6 I 13.5 13.4 13.3 1;,.2 I;,.]
7
1.5.8 15.6 15.5 15.4 15.3
8
18.0 17.9 17.7 17.6 17.5
20.2 20.1 20.0 19.8 19.6
9
10 22.5 22.3 22.2 22.0 21.8
20
45.0 44.7 44.3 44.0 43.7
30
67.5 67.0 66..5 66.0 65.5
40
90.0 8n.3 887 88.0 87.3
50 112.5111.7110.8110.0109.2
130 129 128 127 126
1
2.2
2.2
2.1
2.1
2.1
2
4.3
4.3
4.3
4.2
4.2
6.4
3
6.5
6.4
6.1. 6.3
4
8.7
8.6
8.5
8.5
8.4
10.8 10.8 10.7 10.6 10.5
5
6
13.0 12.9 12.8 12.7 12.6
]5.2 1.';.0 14.9 14.8 14.7
7
8
17.3 17.2 17.1 16.9 16.8
9
19.5 19.4 19.2 19.0 18.9
10 21.7 21.5 21.3 21.2 21.0
20
43.3 43.0 42.7 42.3 42.0
30
6.5.0 64.5 64.0 63.5 63.0
40 86.7 86.0 85.3 84.7 84.0
50 108.3 107.5 106.7 105.8 105.0
125 124 123 122 121
2.1
2.1
2.0
1
2.0
2.0
4.2
4.1
4.1
2
4.1
4.0
3
6.2
6.0
8.3
8.3
8.2
8.1
8.1
10.4 10.3 10.2 10.2 10.1
12.5 12.4 12.3 12.2 12.1
14.6 14.5 14.4 14.2 14.1
8 16.7 16.5 16.4 16.3 16.1
9
18.8 18.6 18.4 18.3 18.2
10 20.8 20.7 20.5 20.3 20.2
20 41.7 41.3 41.0 40.7 40.3
30 62.5 62.0 61.5 61.0 60.5
40 83.3 82.7 82.0 81.3 80.7
50 104.2 103.3 102.5 101. 7 100.8
4
.';
6
7
71
TABLE
0
1
2
3
4
6
6
7
8
9
10
11
12
13
14
16
16
17
18
19
20
21
22
23
24
26
26
27
28
29
30
31
32
33
34
36
36
37
38
39
40
41
42
43
44
45
46
47
48
49
60
51
52
53
54
65
56
57
58
59
60
6cf
III
log sin
9.01923
9.02043
9.02163
d
log tan
9.02162
9.02283
120 9.02404
m
edl
log cot
I
0.97838
m 0.97717
121 0.9759Q
966 1866
I
log eos
999761
9:99760
9.99759
60
59,
58 I
.
1
9.02283 119 9.02525 120 0.97475 9.99757 57 i 2
9.02 402
9.02 645
0.97 355 9. 99 75~ 56 i 3
1118
1211
9.02520119
9.02766 1190.97231
9.02 639
9.
02
88~
0.97 115
118
9.02 757 117 9.03 005 120 0.96 995
119'
9.02874 1189.03124
0.96876
9.02992 117 9.03242 118' 0.96758
119
9.03109 117 9.03361 11810.96639
9.03226 116 9.03479 11810.96521
9.03342 116 9.03597 11710.96403
9.03458 116 9.03714 1180.96286
9.03574 116 9.03832
0.96168
116'1
9.03 690 1 5 9.03 948
0.96 052
9.038051159.04065mO.95935
9.03920 114 9.04181 1160.95819
9.04 034115 9.04
29711161 0.95 703
9.041491113
9.04413 11510.95587
9.04262 !114 9.04528 115 0.95472
9.04376'1149.046431150.953579.99733
9.04490 113 9.04758 115 0.95242
9.04603 112 9.04873 114 0.95 127
9.04 715 113 9.04 987 114 0.95 013
9.04 828 112 9.05 101 113 0.94 899
9.04 940 112 9.05 214 114 0.94 786
9.05 052 112 9.05 328 113 0.94 672
9.05 161 111 9.05441 112 0.94559
9.05275 111 9.05553 113 0.94447
9.05386 111 9.05666 112 0.94334
9.054971110
9.05778 112 0.94222
9.05 607 110 9.05 890 112 0.94 110
9.05717
9.05 827 1110 9.06 113 1,1,1 0.93 887
9.05937 109 9.06221 1111!0.93 776
9.06046 109 9.06 335110! O.93 66~
9.06155 109 9.06445 11110.93555
9.06 264 108 9.06 556 110 0.93 444
9.06372 109 9.06666
0.93 33~
1091
9.06481 108 9.0677~ 1100.93225
9.06 589 107 9.06 885 '109 0.93 115
9.06 696 108 9.06 994 1091 0.93 006
9.06 804 107 9.07 103 108 0.92 897
9.06 911 107 9.07 211 109 0.92 789
9.07 018 106 Q. 07 320 108 0.92 680
9.07 124 107 9.07428 108 0.92572
9.07231 106 9.07536 107 0.92464
9.07337 105 9.07643 108 0.92 357
9.07 442 106 9.07 751 107 0.92 249
9.07548 105 9.07858 106 0.92 142
9.07653 105 9.07964 107 0.92036
9.07758 105 9.08071 106 0.91 929
9.07 863 105 9.08 177 106 0.91 823
9.07 968 104 9.08 283 106 0.91 717
9.08 072 104 9. 08 38~ 106 0.91 611
9.08176 104 9.08495 105 0.91505
9.08280 103 9.08600 105 0.91400
9.08383 103 9.08705 105 0.91295
9.08486 103 9.088101104 0.91 190
9.08589
9.08914
0.91086
logeos I d I log cot ! cd I log tan I
9.99755
9.99 753
9.99 752
9.99751
9.99749
9.99748
9.99747
9.99745
9.99744
9.99742
9.99 741
9.9974044
9.99738
9.99 737
9.99736
9.99734
66
54
53
52
51
60
49
4d
47
46
46
9.99731
9.99730
9.99 728
9.99 727
9.99 726
9.99 724
9.99723
9.99721
9.99720
9.99718
9.99 717
43
42
41
40
39
38
37
36
36
34
33
32
31
30
29
28
9.99 714
9.99713
9.99711
9.99710
9.99 708
9.99707
9.99705
9.99 704
9.99 702
9.99 70 I
9.99 699
9.99 698
9.99696
9.99695
9.99693
9.99 692
9.99690
9.99689
9.99687
9.99 686
9.99 684
9.99 683
9.99681
9.99680
9.99678
9.99677
9.99675
26
25
24
23
22
21
20
19
18
17
16
16
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
tog sin
I
TABLE
III
log sin
d
~76"
I
Prop. Parts
'
6
7
8
9
10
20
30
40
50
1
2
3
6
7
8
9
10
20
30
40
50
121
2.0
4.0
6. (}
120
2.0
119
2..0
118
2.0
6.0
6.0
5.9
4.0
3.~
1~.g
~.~ ~.~
~1~'1
.
12 I 12' 0 11' 9 11' 8
14' 1 14' 0 13' 9 13' 8
16'1 16'015'915'7
18:2 18:0 17:8 17:7
20.2 20.0 19.8 19.7
40.3 40.039.7
39.3
60.5 60.059.5
59.0
80.7 80.0 79.3 78.7
100.8 100.099.298.3
117116116114
2.0 1.9 1.9
3.9 3.9 3.8
5.8 5.8 5.8
1.9
3.8
5.7
~~.~ ~.~ ~.~ ~.~
11'71"611'511'4
13:6 13:5 13:4 13:3
15 6 15 5 15 3 15 2
17: 6 17: 4 17: 2 17: 1
19.5 19.3 19.2 19.0
39.0 38.7 38.3 38.0
58.5 58.0 57.5 57.0
78.0 77.3 76.7 76.0
97.596.795.895.0
113 112
1 \ . 1. 9 1.9
2
3.8 3.7
3 I 5.6 5.6
4,
7.5 7.)
5 i 9.4 9.3
6 11.311.2
7 13.2 13.1
8 15.1 14.9
9 17.0 16.8
10 18.8 18.7
20 37.7 37.3
30 56.5 56.0
40 75.3 74.7
50 94.2 93.3
1
2
3
4
5
6
7
8
9
10
20
30
40
50
4.0
109
1.8
3.6
5.4
7.3
9.1
10.9
12.7
14.5
16.4
18.2
36.3
54.5
72.7
90.8
108
1.8
3.6
5.4
7.2
9.0
10.8
12.6
14.4
16.2
18.0
36.0
54.0
72.0
90.0
111 110
1.8 1.8
3.7 3.7
5.6 5.5
7.4 7.3
9.2 9.2
11.111.0
13.0 12.8
14.8 14.7
16.6 16.5
18.5 18.3
37.0 36.7
55.5 55.0
74.0 73.3
92.5 91.7
107
1.8
3.6
5.4
7.1
8.9
10.7
12.5
14.3
16.0
17.8
35.7
53.5
71.3
89.2
Prop. Parts
106
1.8
3.5
5.3
7.1
8.8
10.6
12.4
14.1
15.9
17.7
35.3
53.0
70.7
88.3
(
70
log tan led,
log cot
970 1870 277/)
log eos
0 9.085891103 9.08914
9.99675
105 0.91086
I 9.08 69~
9.09019
0.90981
9.99674
2 9.08 795 103 9.09 123 104 0.90 877 9.99 672
3 9.08 897 102 9.09 227 104 0.90 773 9.99 670
4 9.08 999 102 9.09 330 103 0.90 670 9.99 669
102
104
6 9.09 101
9.09 434
0.90 566 9.99 667
6 9.09202 101 9.09537 103 0.90463
9.99666
102
103
7 9.09 301
9.09 640
0.90 360 9.99 664
101
102
8 9.094051019.0974210310.902589.99663
9 9.09 506
J00 9.09 845 102 0.90 155 9.99 661
10 9.096061019.099471020.900539.99659
II 9.09 707
9. 10 049
0.89 951 9.99 658
12 9.09 807 100 9. 10 150 10II 0.89 850 9.99 656
13 9.09907 100 9.10252 102
9.99655
99
10110.89748
14 9.10006 1009.10353
9.99653
101 0.89647
16 9.1010Q
9.10451 101 0.89546
9.99 6~1
99
16 9.10205
999.10555110110.894459.99650
17 9. 10 304
9. 10 656 100' 0.89 344 9.99 648
18 9.10402199 98 9.10756 10010.89244
9.99647
(}
19 9. 1 50 I
98 9. J0 856 1001 0.89 144 9.99 645
20 9. 10 599
98 9.10 956 100 0.89 044 9.99 643
21 9.10697,
9.11 056 9910.88941
9.99642
22 9.10 795 98 9.11 155
98
991 0.88 845 9.99 640
23 9.10893
0.88 746 9.99638
97 9.11254
99' 0.88 647 9.99 637
24 9. 10 990
97 9. 11 353
991
26 9.11 087 97 9. 1J 452
0.88 548 9.99 635
26 9. 11 184 97 9. II 551 99 0.88 449 9.99 633
98
27 9. 11 281 96 9. 11 649
98 0.88 351 9.99 632
289.11377
979.11747
980.8825~
9.99630
299.11474969.11845980.881559.99629
30 9. J 1 570 96 9. 1J 943
0.88 057 9.99 627
31 9.11 666 95 9. 12 040 971
98 0.87 960 9.99 625
32 9.11 761 96 9. 12 13§
9.99624
971 0.87862
1~4 9:,19521959:1233219610:876689:99620
135 9.12047
9.99618
95 9.1242§1 97'0.87572
36 9.12142
94 9.1252519610.87475
9.99617
37
~.12236 95 9.12621
9610.87379
9.99615
38
~.1233! 94 9.12717
9610.87283
9.99613
39 9.12425
9610.87187
9.99612
94 9.12813
40 9. 12 519 /93 9. 12 909 95 0.87 091 9.99 610
41 9.12 612 94 9. 13 004 95! 0.86 996 9.99 608
42 9.12706
93 9.13099 I 9510 86901 9.99 60Z
43 9.12799
9.99605
93 9.13194
95 0.86806
44 9.12 892 93 9.13 289 951 0.86 711 9.99 603
45 9. 12 985 93 9. 13 384 941 0.86 616 9.99 601
46 9.13078
939.13478
95'0.86522
9.99600
47 9.13171
94'0.864279.99598
92 9.13573
48 9.13263
9.99596
92 9.13 667 94' 0.86333
49 9.13355
9310.862399.99595
92 9.13761
50 9. 13 447 92 9. 13 854 94 0.86 146 9.99 593
51 9.13539
919.13948
93'0.860529.99591
52 9. 13 630 92 9. 14 041
0.85 959 9.99 589
53 9.13 722 91 9.14 134 93' 0.85 866 9.99 588
54 9. 13 813 91 9. 14 227 931 0.85 773 9.99 586
931
55 9. 13 904 90 9. 14 320 92, 0.85 680 9.99 584
56 9. 13 994 91 9. 14 412
0.85 588 9 99 582
57 9.14 085 90 9. 14 504 93
921 0.85 496 9.99 581
58 9.14175191
9.14597
91 0.85403
9.99579
59 9. 14 266 90 9. 14 688
0.85 312 9.99 577
921 0.85220
60 9.14356
9.14780
9.99575
10 eos
!d
1'120 2620 3520
10 cot!
cd!
10 tan
Prop. Parts
60
106 104 103 102
59
1
1.8 1.7 1.7 1.7
58
2
3.5 3.5 3.4 3.4
57
3
5.2 5.2 5.2 5.1
56
4
7.0 6.9 6.9 6.8
66
5
8.8 8.7 8.6 8.5
54
6 10.5 10.4 10.3 10.2
53
7 12.2 12.1 12.0 11.9
52
8 14.0 13.9 13.7 13.6
51
9 15.8 15.6 15.4 15.3
50 10 17.5 17.3 17.2 17.0
49 20 35.0 34.7 34.3 34.0
48 30 52.5 52.0 51.5 51.0
47 40 70.0 69.3 68.7 68.0
46 50 87.5 86.7 85.8 85.0
45
101 100 99
98
44
1
1.7 1.7 J.6 1.6
43
2
3.4 3.3 3.3 3.3
42
3
5.0 5.0 5.0 4.9
41
4
6.7 6.7 6.6 6.5
40
5
8.4 8.3 8.2 8.2
39
6 10. I 10.0 9.9 9.8
38
7 11.8 11.7 11.6 11.4
37
8 13.5 13.3 13.2 13.1
36
9 15.2 15.0 14.8 14.7
36
10 16.8 16.7 16.5 16.3
34 20 33.7 33.3 33.0 32.7
33 30 50.5 50.0 49.5 49.0
32 40 67.3 66.7 66.065.3
31 50 84.2 83.3 82.5 81.7
30
29
97
96
96
94
28
1 I 1.6 1.6 1.6 1.6
26
3 1 4 8 "1 8 4 8 4 7
4: 6.) 6.4 6.3 6.3
25
8.1 8.0 7.9 7.8
24
~I 9.7 9.6 9.5 9.4
23
7 11.311.211.111.0
22
8 12.9 12.8 12.7 12.5
21
9 14.6 14.4 14.2 14.1
20
10 16.2 16.0 15.8 15.7
19 20
32.3 32.0 31.7 31.3
18
47.0
17 30 48.5 48.047.5
40 64. 7 64.0 63.3 62.7
16
50 80.8 80.0 79.2 78.3
15
14
93
92
91
90
13
1,
1.6 1.5 1.5 1.5
12
2
3.1 3.1 3.0 3.0
11
4.6 4.6 4.6 4.5
3
10
4
6.2 6. I 6. I 6.0
9
5
7.8 7.7 7.6 7.5
8
6
9.3 9.2 9. I 9.0
7
7 10 8 10.7 10.6 10.5
6
8 12.4 12.3 12.1 12.0
9 14.0 13.8 13.6 13.5
5
4 10 15.5 15.3 15.2 15.0
3 20 31.0 30.7 30.3 30.0
2 30 46.5 46.0 45.5 45.0
1 40 162.061.360.760.0
0, 50 77.5 76.7 75.8 75.0
10 sin
tf~o
73
TABLE III
log sin
0
1
2
3
4
6
6
7
8
9
10
11
12
13
14
16
16
17
18
19
20
21
22
23
24
26
26
27
28
29
30
31
32
33
35
36
37
38
39
40
41
42
43
44
46
46
47
48
49
60
51
52
53
54
66
56
57
58
59
60
log tan
d
cd
log cot
0.8522019.99575
92 0.85128
9.99574
91
910.85037
9.99572
910.84942
9.99570
0.84855
9.99568
91
910.84764
9.99562
0.84673
9.99565
90
91 0.84 583 9.99 563
90 0.84 492 9.99 56\
90 0.84 402 9.99 559
89 0.84 312 9.99 557
90 0 84 223 9.99 556
9.99554
89 0.84133
0.84044
9.99552
90
89 0.83 954 9.99 550
9.99548
89 0.83865
88 0.83 776 9.99 542
8910.83688
9.99545
9.99543
88 0.83599
88 0.83 511 9.99 541
88 0.83 42~ 9.99539
880.83335
9.99537
880.83247
9.99535
0.83159
9.99533
87
9.99532
8810.83072
0
82
984
9.99 530
87
87 0.82 897 9.99 528
9.99526
87 0.82810
86 0.82 723 9.99 524
87 0.82 637 9.99 522
86 0.82 550 9.99 520
86 0.82 464 9.99 518
,
0-
:;
0
261
0
361
0
81°
I
Prop. Parts
log cos
60
9.\4780
9.\4356
89 9.14872
59
9.14445
90
9.14535
899.14963
58
9.14624
909.15054
57
56
9.15145
9.14714
89
9.14803
889.15236
66
54
9.14891
89 9.15327
53
9.14 980 89 9. 15 417
52
9.15 069 88 9. 15 508
51
9. 15 157 88 9. 15 598
60
9.15 245 88 9.15 688
49
9. 15 333 88 9. 15 777
48
9.15421
87 9.15867
47
9.15508
88 9.15956
46
9. 15 596 87 9. I 6 04~
45
9.16135
9.15683
87
44
9. 15 770 87 9. 16 224
43
9.15857
87 9.16312
42
9.15944
86 9.16401
41
9. 16 030 86 9. 16 489
40
9.16116
87 9.16577
39
9.16203
869.16665
38
9.16289
859.16753
37
9.16374
86 9.16841
36
9.16460
859.16928
36
9.16 545 86 9.17 016
34
9 16 63 I 85 9. 17 103
33
9.16716
85 9.17190
32
9.16 801 85 9.17 277
31
9.16 886 84 9. 17 363
30
9. 16 97Q 85 9. 17 450
29
9.17 055 84 9.17 536
28
9. 17 139 84 9. 17 622
86 0.82 378 9.99 51 Z
I 8610.82 292 9. ~9? 15 27
9. I 7 22~ I 84 9. 1 ~~08
I
b!
84
0.82 12Q 9.99 511 25
9.17391
83 9. 17 880 85'
24
9.99509
9.17474
84 9. 17 965 86 0.82035
9. 17 558 83 9. 18 051 85 0.81 949 9.99 507 23
9.18136
9.17641
85 0.81 864 9.99 505 22
8' 9.18221
9.17724
85 0.8\ 779 9.99 503 21
8~
9.99501
20
9. 17 807 83 9. 18 306 85 0.81694
8,9.18391
9.17890
84 O.81 602 9 . 99 499 19
9.17973
829.18475
85 0.81 525 9.99 497 18
9.18055
82 9. 18 560 84 0.81 440 9.99 495 17
9.18 137 839.18644
84 0.81 356 9.99 494 16
9. 18 220 87 9. 18 728 84 0.81 272 9.99 492 16
9. 18 302 8\ 9.18812
84 O.81 188 9 . 99 490 14
9. 18 383 82 9. 18 896 83 0.81 104 9.99 488 13
12
9.18465
82 9. 18 979 84 0.81 021 9.99 486 11
9.99484
9.18547
81 9. 19 063 830.80937
9. 18 628 81 9. 19 146 83 0.80 854 9.99 482 10
9
9.99480
9.18709
8\ 9. 19 229 830.80771
8
9.18790
83 0.80 68§ 9.99 478
81 9.19312
7
9.18871
81 9. 19 395 83 0.80 605 9.99 476
6
830.80522
9.99474
9.18952
81 9.19478
6
820.80439
9.99472
9 19 033 80 9.19561
4
9.99470
9.19113
82 0.8035Z
80 9.19643
3
9.19 193 80 9.19725
82 0.80 275 9.99 468
2
9. 19 273 80 9. 19 807 82 0.80 193 9.99 466
1
9.19353
82 O.80 111 9 . 99 464
80 9.19889
0
0.80029
9.99462
9.19971
9.19433
I'll
TABLE III
log sin
980 1880 2780
80
'
1
2
3
4
5
6
7
8
9
10
20
30
40
50
91
92
1.5
1.5
3.0
3.1
4.6
4.6
6.1
6.1
7.6
7.7
9.1
9.2
10.7 10.6
12.3 12.1
13.8 13.6
15.3 15.2
30.7 30.3
46.0 45.5
61. 3 60.7
76.7 75.8
90
1.5
3.0
4.5
6.0
7.5
9.0
10.5
12.0
13.5
15.0
30.0
45.0
60.0
75.0
1
2
3
4
5
6
7
8
9
10
20
30
40
50
89
1.5
3.0
4.4
5.9
7.4
8.9
10.4
11.9
13.4
14.8
29.7
44.5
59.3
74.2
88
1.5
2.9
4.4
5.9
7.3
8.8
10.3
11.7
13.2
14.7
29.3
44.0
58.7
73.3
87
1.4
2.9
4.4
5.8
7.2
8.7
10.2
11.6
13.0
14.5
29.0
43.5
58.0
72.5
0
1
2
3
4
6
6
7
8
9
10
11
12
13
14
16
16
17
18
.
86
1.4
2.8
4
5
6
7
8
9
10
20
30
40
50
1
2
3
4
5
6
7
8
9
10
20
30
40
50
:
1
5.7
7.2
8.6
10.0
11.5
12.9
14.3
28.7
43.0
57.3
71.7
5.7
7.1
8.5
9.9
11.3
12.8
14.2
28.3
42.5
56.7
70.8
5.6
7.0
8.4
9.8
11.2
12.6
14.0
28.0
42.0
56.0
70.0
f3
1.4
2.8
4.2
5.5
6.9
8.3
9.7
11. 1
12.4
13.8
27.7
41.5
55 3
69.2
82
1.4
2.7
4.1
5.5
6.8
8.2
9.6
10.9
12.3
13.7
27.3
41.0
54.7
68.3
81
1.4
2.7
4.0
5.4
6.8
8.1
9.4
10.8
12.2
13.5
27.0
40.5
54.0
67.5
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
~4
35
36
37
38
39
40
41
42
43
44
46
46
47
48
49
50
51
52
53
54
66
56
57
58
59
9.19433
9.19513
9. 19 592
9.19672
9. 19 751
9.19 830
9. 19 909
9. 19 988
9.20067
9.20145
9.20223
9.20 302
9.20 380
9.20 458
9.20535
9.20613
9.20 69 I
9.20768
74
I log tan
led
i
80 9.19971
79 9.20053
9.20 134
80
799.20216
79 9.20 297
79 9.20 378
79 9.20 459
79 9.20 540
78 9.20621
78 9.20701
799.20782
862
78 9.20
0.20942'
~~9.21022
78 9.21 102
78 9.21182
77 9.21 261
77 9.21341
log cot
I
990 1890 2790
log cos
0.80029
9.99462
82 0.79947
9.99460
81 0.79 866 9.99 458
82
810.79784
9.99456
0.79 703 9.99 454
81
0.79 622 9.99 452
81 0.79 541 9.99 450
81 O.79 460 9.99 448
81 0.79379
9.99446
80 0.79299
9.99444
81
8010.792189.9944260
0.79 138 9.99 440
80 10.79 058 9.99438
0.78978
9.99436
~~!
0.78 898 9.99 434
801
0.78818
9.99432
79 0.78 739 9.99 429
80' 0.78659
9.99427
9.20 845
9.21 420 79 I 0.78 580 9 99 425
9.20 922
~~9.21499
0.78501
9:99423
~~i
9.20999
9.21578
7910.78422
9.99421
77
9.21076
7910.783'/3
).99419
77 9.21657
9.21 153 76 9.21736
7810.78264
9.99417
0.78186
9.99415
9.21229
77 9.21814
9.21 306 76 9.21 893 79 0.78 107 9.99 413
78
0.78029
9.99411
9.21382
76 9.21971
78 0.77 951 9.99409
9.21458
9.22049
0.77872
9.99407
9.21 534
~~9.22127
~~i
9.21 610
9.22 205
: 0.77 795 9.99 404
0.77 717 9.99 402
9.21 685
~~9.22283
~~I
0.77 639 9.99 400
9.21 761 75 9.22 361
0.77 562 9.99 398
9.21 836 76 9.22 438 771
78
9.21 912
9.22 516
O. 77 484 9.99 396
0.77407 9.99394
9.21 9871
~~9.22593
~~i
i
9.22062'
-22670. -~ 0 77 330 9 99392
9.22 1371 74 9.22 7471 77' 0.77 253 999390
9.22824
7710.77176
9.99388
9.22211
75 9.22901
760.77099
9.99385
9.22286
75
0.77 023 9.99 383
9.22 977
9.22 361
77 0.76946
74 9.23054
9.99381
9.22435
76
74
0.76870
9.99379
9.23 130
9.22509
76 0.76794
74
9.99377
9.22583
749.23206
77 0.76717
9.99375
9.23283
9.22657
74
76
9.22731
76,0.76641
9.99372
74 9.23352
0.76565
9.99370
9.22805
9.23435
73
75
9.22878
9.23510
7610.76490
9.99368
74
9.22 952
9.23 586 751 0.76 414 9.99 366
73
9.23 025
9.23 661
0.76 339 9.99 364
9.23 098 73 9.23 737 76' 0.76 263 9.99 362
'
9.23171
9.23 812
0.76188
9.99359
~~I
9.23 244
9.23 887
0.76 113 9.99 357
73 9.23962" 75' 0.76038
9.23317
9.99355
9.23 390
9.24 037
0.75 963 9.99 353
~~I
~i 9.24112
9.23 46~
0.75888
9.99351
73
74' 0.75 814 9.99348
9.23535
72 9.24186
75'
9.23607
9.24261
7410.75739
9.99346
9.23 679 72 9.24 335 751 0.75 665 9.99 344
73
9.23752
9.24410,
7410.75590
9.99342
71 9.24484 1740.75516
9.23823
9.99340
72
9.23895
72 9.24558174'0.754429.99337
n
60 9.23 967
log cos
Pro. Parts
90
d
I
Prop. Parts
I
60
59
58
57
56
66
54
53
52
51
49
48
47
46
46
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
6
4
3
2
1
9.24 632
0.75 368 9.99 335 0
log cot ! cd! log tan I log sin I
'I
1
2
3
4
5
6
7
8
9
10
20
30
40
50
78
79
80
1.3 1.3 1.3
2.7 2.6 2.6
4.0 4.0 3.9
5.3 5.3 5.2
6.7 6.6 6.5
8.0 7.9 7.8
9.3 9.2 9. I
10.7 10.5 10.4
12.011.811.711.6
13.3 13.2 13.0
26.7 26.3 26.0
40.0 39.5 39.0
53.352.7
52.0
66.7 65.8 65.0
12.8
25.7
38.5
51.3
64.2
1
2
3
4
5
6
7
8
9
10'
20!
30 I
76
1.3
2.5
3.8
5. I
6.3
7.6
8.9
10.1
11.4
12.7
25 3
38.0
74
1.2
2.5
3.7
4.9
6.2
7.4
8.6
9.9
11.1
12.3
24 7
37.0
73
1.2
2.4
3.6
4.9
6.1
7.3
8.5
9.7
11.0
12.2
24.3
36.5
40
50
76
1.2
2.5
3.8
5.0
6.2
7.5
8.8
10.0
11 2
12.5
25 0
37.5
77
1.3
2.6
3.8
5.1
6.4
7.7
9.0
10.3
50.7 50.0 49.3 48.7
I
63.3 62.5 61.7 60.8
72
71
1
1.2 1.2
2.4 2.4
2
3.6 3.6
3
4
4.8 4.7
6.0 5.9
5
6
7.2 7.1
8.4 8.3
7
9.6
9.5
8
9 10.8 10.6
10 12.0 11.8
20 12..023.7
30 36. 0 35.5
40 48.0 47.3
50 60.0 59.2
3
0.0
O. I
0.2
0.2
0.2
0.3
0.4
0.4
0.4
0.5
1.0
1.5
2.0
2.5
2
0.0
0.1
0.1
O. I
0.2
0.2
0.2
0.3
0.3
0.3
0.7
1.0
1.3
1.7
I
!d
1700 2600 360.
Prop.
Parts
75
l
TABLE
100
III
log sin
0
I
2
3
4
5
6
7
8
9
10
II
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
136
37
38
39
40
41
42
43
44
45
46
47
148
.49
50
51
52
53
54
55
56
57
58
59
60
1
9.23967
9.24 039
9.24110
9.24181
9.24253
9.24321
9.24395
9.24466
9.24536
9.24607
9.24677
9.24748
9.24818
9.24888
9.24 958
9.25 028
9.25098
9.25168
9.25237
9.25 307
9.25376
9.25445
9.25514
9.25583
9.25652
9.25721
9.25 790
9.25 858
9.25 927
9.25995
9. 26 063
9.26 131
9.26199
9.26 267
9.26 335
9.26403!
9.26470'
9.26 538
9.26605
9.26 672
9.26 739
9.26 806
9.26873
9.26940
9.27 007
9.27 073
9.27 140
9.27 206
9.27273
9.27 339
9.27405
9.27 471
9.27 537
9.27602
9.27668
9.27 734
9.27799
9.27864
9.27930
927995
9.28 060
I
\690
log cos
d
log tan
c di
72 9.24632
71 9.24 706
71 9.24779
72 9.24853
71 9.2~ 926
71 9.25000
71 9.25073
70 9.25146
71 9.25219
70 9.25292
9.25365
71
709."25437
.
1
70 9.25510
70 9.25 58~
70 9.25 655
70 9.25 727
70 9.25799
69 9.25871
70 9.25942
69 9.26 015
69 9.26086
69 9.26158
9.26229
69
699.26301
69 9.26372
699.26443
68 9.26 511
69 9.26 585
68 9.26 655
68 9.26726
68 9.26 797
68 9.26 867
68 9.26937
9.27 008
~~ 9.27 078
67 9.27148!
9.27218'
~~ 9.27 288
67 9.27357
67 9.27 427
67 9.27 496
67 9.27 566
67 9.27635
67 9.27704
66 9.27 773
67 9.27 842
66 9.27 911
67 9.27 980
66 9.28049
66 9.28 117
66 9.28186
66 9. 28 254
65 9.28 323
66 9.28391
66 9.28459
65
65
66
65
65
Id I
2590 3490
9.28 52Z
9.28595
9.28662
9.28730
9.28798
9.28865,
log cot!
log cot
1000 1900 2800
74 0.75368
73 0.75 294
74 0.75221
73 0.75147
74 0.75074
73 0.75000
73 0.74927
73 0.74854
73 0.74781
73 0.74708
0.74635
72
730.74563
1
9.99 335
9.99 333
9. 99 331
9.99 328
9.99 326
9.99 324
9.99322
9.99319
9.9931Z
9. 99 31 5
9. 99 31 3
9.99 310
9. 99 308
72 0.74490
9. 99 306
73 0.74418
72 0.74 345 9.99 304
9. 99 30 I
72 0.74273
9.99 299
72 0.74201
9.99297
77 0.74129
9.99294
72 0.74057
71 0.73 985 9. 99 292
9.99 290
72 0.73914
9.99 288
71 0.73842
9.99 285
72 0.73771
710.73699
9.99283
9. 99 281
71 0.73628
710.73557
9.99 278
71 0.73 486 9.99 276
70 0.73 41 ~ 9.99274
71 0.73 345 9.99271
9.99 269
71 0.73274
70 0.73 203 9.99 267
70 0.73 133 9.99 264
9.99 262
71 0.73063
0.72 992 9.99 260 I
70
7010.72 922
1
I
I
1
7010.72 852
10.72 782
~~ 0.72 712
70 0.72 643
69 0.72 573
70 0.72 504
69 O. 72 431
69 0.72 365
69 0.72 296
69 O. 72 227
69 0.72 158
69 O. 72 089
69 0.72 020
68 0.71951
69 0.71 883
68 0.71814
69 O. 71 746
68 0.71 677
68 0.71609
68 0.71541
68 0.71 473
67 0.71405
68 0.71338
68 0.71270
67 0.71202
0.71 135
c d
I
9.99257
9. 99 255
9.99 252
9.99 250
9.99 248
9.99245
9.99243
9. 99 241
9.99 238
9. 99 236
9.99 233
9.99231
9.99 229
9.99 226
9.99 224
9.99 221
9 . 99 219
9. 99 21 7
9. 99 214
9. 99 21 2
9. 99 209
9.99 207
9. 99 204
9.99 202
9. 99 200
9.99197
9.99195
log sin
log tan
79°
J
d
106 cos
2 60
2 59
3 58
2 57
2 56
2 55
3 54
2 53
2 52
2 51
3 50
2 49
2 48
2 47
3 46
2 45
2 44
3 43
2 42
2 41
2 40
3 39
2 38
2 37
3 36
2 35
2 34
3 33
2 32
2 31
3 30
2 29
2 28
3
I ~
'3
2
2
3
2
2
3
2
2
2
2
3
2
3
2
2
3
2
3
2
3
2
2
3
2
I
d
27
26
25
24
23
22
21
20
19
18
I7
16
15
14
13
12
II
10
9
8
7
6
5
4
3
2
I
0
74
1
1.2
2
2.5
3
3.7
4
4.9
56.26.16.0
6
7.4
7
8.6
8
9.9
9 11.1
10 12.3
20 24. 7
30 37.0
40 49. 3
50 61 . 7
73
I. 2
2.4
3.6
4.9
72
1.2
2.4
3.6
4.8
7.3
8.5
9.7
11.0
12.2
24. 3
36.5
48. 7
60. 8
7.2
8.4
9.6
10.8
12.0
24.0
36.0
48 . 0
60. 0
71
70
1
1.2
1.2
2
2.4
2.3
3
3.6
3.5
4
4.7
4.7
5
5.9
5.8
67.17.06.9
7
8.3
8.2
8
9.5
9.3
9 10.6 10.5
I (j 11. 8 ! 1.7
20 "121.7
23.3
30
!
35.5
0
I
2
3
4
5
6
7
8
9
10
II
12
13
14
115
'16
17
18
19
20
21
22
23
24
25
126
27
28
29
30
31
69
1.2
2.3
3.4
4.6
5.8
1
8.0
9.2
10.4
11.5
23.0
~2
.)3
35.0 34.5
I
1
67
1.1
2.2
3.4
4.5
5.6
6.7
7.8
66
1.1
2.2
3.3
4.4
5.5
6.6
7.7
10.0
11.2
22.3
33.5
44.7
55.8
9.9
11.0
22.0
33.0
44.0
55.0
Prop. Parts
76
I
log sin
I
9.2806Q
9.28125
9.28 190
9.28 254
9.28 319
9.28 384
9.28448
9.28 512
9.28 577
9.28 64!
9.28 705
9.28 769
9.28 833
9.28 896
9.28 960
9. 2? 024
9.29 087
9.29 150
9.29 214
9.29 277
9.29 34D
9.29 403
9.29 466
9.29 529
9.29591
9.29654
9.29716
9.29 779
9.29 841
9.29 903
9.29 966
9.30 028
d
I
log tan
led
65 9.28865
65 9.28933
64 9.29 000
65 9.29 067
65 9.29 134
64 9.29 20 I
64 9.2926~
65 9.29 335
64 9.29 402
64 9. 29 46~
64 9.29 535
64 9.29 60 I
63 9.29 668
64 9.29 734
64 9.29 800
63 9.29 866
63 9.29 932
64 9.29 998
63 9.30 064
63 9.30 130
63 9.30 195
63 9.30 261
63 9.30 326
62 9.30 391
63 9.30457
62 9.30522
63 9.30587
62 9.30 652
62 9.30 717
63 9.30 782
62 9.30 846
62 9.30 911
9.30 O?O ,
61 9.30 975
9.30 bl
9.31040
0 ..,(1110 'l 62 n
'f"
"'
OL'
36
9.30 336
37
9.30398
38
39
40
41
42
9.30 459
9.30521
9.30 582
9.30643
9.30 704
43
44
45
46
47
48
49
9.30765
9.30826
9.30 887
9.30947
9.31008
9.31 068
9.3 I 129
62 9.31 233
61 9.31 297
62
61
61
61
61
9.31 361
9.31425
9.31 489
9.31552
9.31 616
61
61
60
61
60
61
60
9.31679
9.31 743
9.31 806
9.31 870
9.31933
9.31 996
9.32 059
60
60
59
60
60
59
60
9.32 311
9.32373
9.32436
9.32 498
9.32561
9.32 623
9.32 685
9.32747
I
log cot
I
/
/
!
/
v
o'tl
65! 0.68 832
64 0.68 767
[
64 0.68703
64
64
63
64
63
64
63
64
63
63
63
63
1
0.68 639
0.68575
0.68 51I
0.68448
0.68 384
0.68321
0.68257
0.68 194
0.68 130
0.68067
O.68 004
0.67 941
50 9.31 1~9 61 9.32 122 63 O.67 87~
51 9.31 250 60 9.32 185 63 0.67 815
52 9.31 310 60 9.32 248 63 0.67 752
53 9.31 370
..
54 9.31430
55 9.31490
56 9.3 I 549
57 9.31609
58 9.31 669
59 9.31 728
60 9.31788
1680 2580 3480
62
63
62
63
62
62
62
)
0.67 689
0.67627
0.67564
O. 67 502
0.67439
0.67 37Z
0.67 315
0 67253
9.99
78°
I3
112
/.99109
9.99 106
9.99 104
9.99101
9. 99 099
9. 99 096
9.99 093
9. 99 091
9. 99 088
9.99 086
9. 99 083
9. 99 080
9.99078
9.99075
9.99 072'
9.99 070
9.99 067
9. 99 064
9. 99 062
9.99 059
9.99 056
9.99 054
9.99 051
9. 99 048
9. 99 046
9. 99 043
9 .99 040
Prop. Parts
d
1
/
1
1010 19102810
I J
log cos
68 0.71 135 9.99195
9. 99 192 3
67 0.71067
2
67 0.71 000 9.99 190 3
67 0.70 933 9.9918Z
2
67 0.70 866 9.99 185 3
O.70 799 9.99 182
67
670.70732
9.99 180 2
3
0.70
665
9.9917Z
67
2
0.70
598
9.99
175
66
3
67 0.70 532 9. 99 I 72 2
0.70
465
9.
99
170
66
3
67 0.70 399 9.99 16Z 2
0.70
332
9.99
165
66
3
66 0.70 266 9.99162
2
O.
70
200
9.
99
160
66
3
66 0.70 134 9.9915Z
2
66 O.70 068 9.99 155 3
66 O.70 002 9.99 1~2 2
66 O.69 936 9.99150
3
65 0.69 870 9.99 147 2
0.69
805
9.99145
66
3
65 ' 0.69 739 9. 99 142 2
O.69
674
9.
99
140
65
3
66 0.69 609 9.9913Z '237
0.69543
9.
99
135
65
3
9. 99 I 32 2
65 0.69478
9. 99 130 3
65 0.69413
65 0.69 348 9.99127
3
65 0.69 283 9.99 124 2
64 0.69 218 9. 99 122 3
65 0.69 154 9. 99 I 19 2
64 0.69 082 9. 99 117 3
65, 0.69 025 9. 99 I 14 I 2
,
i. 6410.68900
n "ooor
.
35 9.30 275 61 9.31 168
40 ' 47. 3 46. 7 46. 0
50 59.2 58.3 57.5
68
1
1.1
2 2.3
3
3.4
4
4.5
5 5.7
6 6.8
7
7.9
89.18.98.8
9 10.2
10 11.3
20 22. 7
30 34.0
40 45.3
50 56.7
lr
TABLE III
,
Prop. Parts
1
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
4I
40
39
38
36
35
34
33
32
31
30
29
28
27
13
2 25
3 24
2 23
3 22
3 2J
2 20
3 19
2 18
3 17
3 16
2 15
3 14
3 13
2 12
3 11
3 10
2
9
3
8
3
7
6
2
3
5
3
4
2
3
3
2
3
I
0
1
2
3
4
5
6
7
8
9
10
20
30
40
50
I
I
65
I. I
2.2
3.2
4 3
5 4
6.5
7.6
8.7
9 8
10.8
21.7
32.5
43.3
54. 2
4
5
6
7
8
9
10
62
1.0
2.1
~.I
4. I
5.2
6.2
7.2
8.3
9.3
[10.3
20
20.7
1
2
.
30 131.0
40 41.3
50 51.7
59
1
1. 0
2
2.0
3
3.0
4
3.9
5
4.9
6
5.9
7
6.9
8
7.9
9
8.8
10
9.8
20 19.7
30 29.5
40 39 . 3
50 49. 2
/
64
1.1
2. I
3.2
4.3
5.3
6.4
7.5
8.5
9.6
10.7
21.3
32.0
42.7
53.3
63
1.0
2.1
3.2
4.2
5.2
6.3
7.4
8.4
9.4
10.5
21.0
31.5
42.0
52.5
61
60
1.0
1.0
2.0
2.0
3.0
3.0
4. I
4.0
5. I
5.0
6. I
6.0
7. I
7.0
8. I
8.0
9.2
9.0
10.2 10.0
20.;
20.0
30.5 30.v
40.7 40.0
50.8
50.0
3
0.0
O. I
0.2
0.2
0.2
0.3
0.4
0.4
0.4
0.5
1.0
1.5
2.0
2.5
2
0.0
O.I
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.3
0.7
1.0
1.3
1.7
77
log sin
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
d
log tan
log cot
cd
0.67253
9.32747
59 9.32 810 63 0.67 190
60 9.32872 62 0.67128
59 9.32 933 61 0.67 06Z
62
59
9.32 995 62 0.67 005
59
599.33057
620.66943
9.33119 61 0.66881
59
0.66 820
9.33
180
59
62
9.33242 6110.66758
58 9.33 30~ 6210.66697
59
599.33365
610.66635
0.66574
9.33426
58 9.33 487 61 0.66 513
58 9.33 548 61' 0.66 452
61
59
0.66391
58 9.33609 61
589.33670
610.66330
0.66269
58 9.33731 61 0.66 208
58 9.33 792 61
58 9.33853 60' 0.66 147
0.66 087
58 9.33 913
61[ 0.66026
9.33974
60
58
57 9.34031 61 0.65966
9.34095
0.65 90~
58
9.33133 57 9.3415'160 6010.6584;
9.33190 58 9.34215 61 0.65785
9.33248 579.34276
600.65724
9.33305
57 9.34336 60 0.65664
9.33 362 58 9.34 396 60 0.65 604
9.33420 579.34456
600.65544
9.33477 57 9.34516 6010.65484
9.33534 57 9.34576 59 0.65421
9.33591 56 9.34 63~ 6010.65365
9.31788
9.31 847
9.31907
9.31 96§
9.32 025
9.32084
9.32143
9.32 202
9.32261
9.32319
9.32378
9.32437
9.32 495
9.32 553
9.32612
9.32670
9.32728
9.32 786
9.32844
9.32 902
9.32960
9.33018
9.33075
log cos
d
9.99040
9.99 038
9.99035
9.99 032
9.99 030
9.99027
9.99024
9.99 022
9.99,,19
9.99016
9.99013
9.99011
9.99 008
9.99 005
9.99002
9.99000
9.98997
9.98 994
9.98991
9.98 989
9.98986
9.98983
9.98980
9.98971
9.98975
9.98972
9.98969
9.98 967
9.98964
9.98961
9.98958
9.98955
2
3
3
2
3
3
2
3
3
3
2
3
3
3
2
3
3
3
2
3
3
3
2
3
3
3
2
3
3
3
3
2
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
4\
40
39
38
37
36
35
34
33
32
31
30
29
57 9.34 6?~ 60~0.65 305 9.98~. 3 28
27
L
~34 9~
57 9.34814: 60,0.65186 9 98947: 3 26
35 9.338181 56 9.34874 i 59' 0.65 126 9.989441 3 25
24
32 133647
36 9.33874 57 9.34933 5910.65067 9.989411
37 9.33 931 56 9.34 992 5910.65 008 9.98 938 i
38 9.33 987 56 9.35 051 60 0.64 949 9.98 936
399.34043579.35111590.648899.989333
40 9.34100 56 9.35170 59 0.64830 9.98930
41 9.34 156 56 9.35 229 59 0.64 771 9.98 927
42 9.34 212 56 9.35 288 59 0.64 712 9.98 924
43 9.34268 56 9.35347 58 0.64 65J 9.98921
44 9.34 324 56 9.35 405 59 0.64 595 9.98 919
9.98916
45 9. 34 380 56 9.35464 590.64536
4) 9. 34 436 55 9.35 523 58 0.64 477 9.98 913
0.64
419
9.98 910
47 9. 34 491 56 9.35 581 59
48 9.34547 55 9.35 640 58 0.64 360 9.98 907
49 9.34 602 56 9.35 698 59 0.64 302 9.98 904
50 9.34 658 55 9.35 75Z 58 0.64243 9.98901
51 9.34713 56 9.35 815 58 0.64 185 9.98 898
52 9. 34 769 55 9.35873 58 0.64127 9.98896
53 9. 34 824 55 9.35 931 58 0.64 069 9.98 893
54 9.34879 55 9.35989 58 0.64011 9.98890
55 9.34934 559.36047
58 0.6395J 9.98887
56 9.34 989 55 9.36 105 58 0.63 895 9.98 884
9.36
163
57 9.35 044 55
58 0.63 837 9.98 881
58 9.35 099 55 9.36 221 58 0.63 779 9.98 878
59 9.35154 55j 9.36 279 5710.63721 9.98875
60 9.35 209
9.36 336
0.63 664 9.98 872
~10..
cot! c d 10.. tan I 101':sin
3
3
3
2
3
3
3
3
3
3
3
2
3
3
3
3
3
3
3
3
23
22
21
20
19
18
17
16
15
14
13
12
II
10
9
8
7
6
5
4
2
2
1
0
Id I ' I
I
1670 2570 3470
3
2
3
77'"
1
2
3
4
5
6
7
8
9
10
20
30
40
50
63
1.0
2.1
3.2
4.2
5.2
6.3
7.4
8.4
9.4
10.5
21.0
31.5
42.0
52.5
1
2
3
4
5
6
7
8
60
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
,9
, 9.0
'D 10.0
i-zu-.
,
62
61
1.0
1.0
2.1
2.0
3.1
3.0
4.1
4.1
5.2
5. I
6.2
6.1
7 2 7.1
8.3
8.1
9.3
9.2
10.3 10.2
20.7 20.3
31.0 30.5
41.3 40.7
51.7 50.8
59
1.0
2.0
3.0
3.9
4.9
5.9
6.9
7.9
8.8
9.8
58
1.0
1.9
2.9
3.9
4.8
5.8
6.8
7.7
8.7
9.7
30! 30.0 29. ') 29.0
40 40.0 39.3 38.7
50 50. 0 49. 2 48. 3
8,-
I
1
2
3
4
5
6
7
8
9
10
20
30
40
50
57
1.0
1.9
2.8
3.8
4.8
5.7
6.6
7.6
8.6
9.5
: 19.0
128.5
38.0
47.5
56
0.9
1.9
2.8
3.7
4.7
5.6
6.5
7.S
8.4
9.3
18.7
28.0
37.3
46.7
55
0.9
1.8
2.8
3.7
4.6
5.5
6.4
7.3
8.2
9.2
18.3
27.5
36.7
45.8
log sin
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
\9
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
9.35209
9.35263
9.35318
9.35 373
9.35 427
9.35481
9.35536
9.35590
9.35644
9.35 698
9.35752
9.35 806
9.35860
9.35914
9.35 968
9.36 022
9.36075
9.36129
9. 36 182
9.36236
9.36 289
9.36342
9.36 395
9. 36 449
9.36 50~
9.36 555
9.36608
9. 36 660
9.36713
9.36766
9.36 819
9.36871
35
9.3708\
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
9.37133
9.37185
9.37 237
9.37 289
9.37341
9.37393!
78
~~~-
log tan
cd
1
. .H VZ'O't
53
52
'J..JV
L.JI
9.38313!
I
log cos
d
log cot
log cos
log cot
I 0.63 664
0.63 606
~~l
0.63548
57 0.63491
57i
0.63 434
58
0.63376
57 O.63 319
57 0.63 26~
57 0.63205
57 0.63148
57
0.63091
57 0.63 034
57 0.62977
57 0.62920
57 0.62863
56
0.62807
57 0.62750
56 0.62694
57 0.62637
56 0.62581
57
0.62524
56 0.62468
56 0.62412
56 0.62356
56 0.62300
56
0.62244
56 0.62188
56 0.62132
56 0.62 076
56
O.62 02~
55
0.61 965
56 0.61909
56 0.61853
55 0.61 798
'is V.VI I'r.J
9.98 872
9. 98 869
9.98 867
9. 98 864
9. 98 861
9.98 85§
9. 98 855
9. 98 852
9. 98 849
9. 98 846
9. 98 843
9. 98 840
9.98837
9.98 834
9.98 831
9.98 828
9.98 825
9.98 822
9. 98 819
9. 98 816
9. 98 813
9. 98 810
9.98 807
9. 98 804
9.98 801
9. 98 798
9.98 795
9.98 792
9.98789
9.98 786
9.98 783
9.98780
9.98 777
9.98 774
5"10.61687
9.98
7.7V
. 5f).
9.38 368 i
52 9.38423
52 9.38479
52
52 9.38 534
9.38 589
52 9.38644
52 9.38699
9.37445
52
9.37497 52 9. 38 754
9.37549
51 9.38808
9.37600 52 9.38 863
9.37652
9.38 918
9.37 70J 51 9.38 972
52
9.37755
5\ 9.39 027
9.37806152 9.39082
9.37858
9. 39 136
9.37909 51 9.3919Q
51
9.37960
9.39245
9. 38 011 51 9.39 299
51
9. 38 062
51 9.39353
9. 38 113
9.39 407
9. 38 164 51 9.39461
51
9. 38 215
9.39515
9.38 266 5 I 9.39569
51
9.38 317
51 9.39623
9.38368
9. 39 677
I
Prop. Parts
d
54 9.36 336
55 9.36 394
55 9.36452
54 9.36509
54 9.36 566
9.36624
55
54 9. 36 68\
54 9.36738
54 9.36795
54 9.36852
9.36909
54
966
54 9.36
9.37023
54 9.37080
54
54 9.37137
9.37193
53
54 9.37250
53 9.37306
54 9.37363
53 9.37419
9.37476
53 9.37532
53 9.37588
54 9.37644
53 9.37700
53
9.37756
53 9.37812
52 9.37868
53 9.37 924
53
9.37 980
53
9.38 035
52 9.38091
53
9.36 924 52 9.38147
9.36976: <;7 9.38 202
y
1030 1930 283°
138
TABLE III
102~ 192~ 282°
Prop. Parts
12°
TABLE III
'.
76~!
9.9876)'
9. 98 762
9.98759
9.98756
9. 98 7~3
9. 98 750
9. 98 746
9.98743
9. 98 740
9.98737
9.98 734
9.98 731
9.98 72§
9.98 725
9.98 722
9.98 719
9.98 715
9.98 712
9. 98 709
9. 98 706
9.98 703
9. 98 700
9.98 697
9.98 694
I 0.60 323 9.98 690
5510.61 632
5610.61577
5510.61521
0.61 466
55
0.61 411
55
5510.61356
0.6130\
55 0.61 246
54
5510.61 192
55, 0.61 137
0.61 082
54' 0.61 028
55 0.60 973
55
5410.60918
0.60 864
54 0.60810
55
5410.60755
54! 0.60 701
54,0.60647
5410.60 593
54: 0.60532
5410.60485
0.60431
54
54,0.60377
cd
log
~an
I
log
sin
I
d
3
2
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
3
3
3
3
3
3
3
3
3
4
3
3
3
3
3
3
3
4
d
Prop. Parts
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
LV
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
'
57
56
55
1.00.90.9
1.91.91.8
2.8
2.8
2.8
3.8
3.7
3.7
4.8
4.7
4.6
5.7
5.6
5.5
6.6
6.5
6.4
7.6
7.5
7.3
8.6
8.4
8.2
9.5
9.3
9.2
19.0 18.7 18.3
28.5 28.0 27.5
38. 0 37.3 36. 7
47.5 46.7 45.8
1
2
3
4
5
6
7
8
9
10
20
30
40
50
54
53
0.9
0.9
1.81.81.7
2.7
2.6
3.6
3.5
4.5
4.4
5.4
5.3
6.36.26.1
7.27.16.9
8.18.07.8
8.8
.?'~
,v. V -1-1-:-1--".
1
2
3
4
5
6
7
8
9
!O
LV
30,
27.0
26.5
52
0.9
2.6
3.5
4.3
5.2
,~.!
J
26.0
40 36.0 35.3 34.7
50
1
2
3
4
5
6
7
8
9
10
20
30
40
50
I
45.0
44.2
43.3
51 4 3
0.8 0.1 0.0
1.70.10.10.1
2.60.2 0.2
3.4 0.3 0.2
4.2 0.3 0.2
5.1 0.4 0.3
6.00.50.40.2
6.8 0.5 0.4
7.6 0.6 0.4
8.50.7 0.5
17.0 1.3 1.0
25.5 2.0 1.5
34.0 2. 7 2. 0
42.5 3.3 2.5
Prop. Parts
2
0.0
0.1
0.1
0.2
0.2
0.3
0.3
0.3
0.7
1.0
1. 3
1.7
[
log sin
log tan
d
I cd
1
1
09.38368609.3967754'0.60323
I 9.38 418 5\ 9.39 731 54 0.60 269
0.60 215
2 9.38 469 50 9.39 785
0.60 162
3 9.38 519 5\ 9. 39 838 53'
54'
9.39892
53,0.60108
4 9.38570
50
6 9.38 620 50 9.39 945 54' 0.60 055
6 9.38 670 51 9.39 999 53 0.60 001
0.59948
7 9.38721
50 9.40052
54'
53,0.59894
8 9.38771
50 9.40106
9 9.38 821 50 9.40 159 53 0.59 841
10 9.38871
509.40212
540.59788
11 9.38 921 50 9.40 266 53 0.59 734
12 9.38971
53 0.5968\
50 9.40319
0.59628
13 9.39021
50 9.40372
53' 0.59575
14 9.39071
53
50 9.40425
0.59522
16 9.39121
53
49 9.40478
16 9.39 170 50 9.40 531 53 0.59 469
17 9.39 220 50 9.40 584 521 0.59 416
0.59 364
18 9.39 270 49 9.40 636
53'
19 9.39 319 50 9.40 689 53 ' 0.59 311
0.59258
20 9.39369
53
49 9.40742
2\ 9.39 418 49 9.40 795 52 0.59 205
22 9.39467
530.59153
50 9.40847
23 9.39 517 49 9.40 900 52 0.59 100
0.59 048
24 9. 39 56~ 49 9.40 952
531
0.58995
26 9.39615
52 0.58943
49 9.41005
26 9.39664
52 0.58891
49 9.41057
27 9.39713
52
49 9.41109
28 9.39 762 49 9.41 161 53 0.58 839
0.58786
29 9.39811
52
49 9.412\4
30 9.39 860 49 9.41 266 52 0.58 734
31 9.39 909 49 9.41 318 52 0.58 682
32 9.39 958 48 9.41 370 52 0.58 630
0.58578
33 9~Q Q?~ 49 9.41422
52 I 0 58 52/\
O 'II! 11'1'
0 11 471
I
I - 0.58474
9.41526
36 9.40103
49
~~I 0.58 422
36 9.40 152 48 9.41 578
511
9.41
629
52,0.58371
37 9.40200
49
38 9.40 249 48 9.41 681 521 0.58 319
39 9.40 297 49 9.41 733 51 0.58 267
40 9.40 346 48 9.41 784 52 0.58 216
0.58164
41 9.40394
51 0.58 113
48 9.41836
9.41887
42 9.40442
52 0.58061
48
43 9.40490
51
48 9.41939.
44 9.40 538 48 9.41 990 ' 51 0.58 010
1
1
46
46
log cas
9.98 690
9.98 687
9.98 684
9.98 681
9. 98 67~
9.98 675
9.98 671
9 . 98 668
9.98 665
9 98 662
9.98 659
9.98 656
9.98 652
9.98 649
9.98 646
9.98 643
9 .98 640
9.98 636
9.98633
9.98 630
9.98627
9.98623
9.98 620
9.98617
9.98 614
9 .98 610
9.98 607
9. 98 604
9.98601
9.98597
9.98 594
9.98 591
9.98 588
9.98 584
9 98 581
9.98 578
9.98574
9.98571,
9.98 56~
9.98 565
9.98 561
9. 98 55~
9.98555
9.98551
9. 98 54~
9.40586
9.42041
0.57959 9.98 545
9.40 634 :~ 9.42 093 1 n' [0.57 907 9.98 541
0.5785§
9.9853~
9.40682
51 0.57805
48 9.42144
9.98535
9.40730
51 0.57754
48 9.42195
9.98 531
9.42246
9.40778
1
47
48
49
47
9.42297
60 9.40825
48
51 9.40873
48 9.42348
52 9.40921
47 9.42399
53 9.40968
48 9.42450
54 9.41016
47 9.42501
66 9.41063
48 942552
56 9.41 111 47 9.42 603
57 9.41 158 47 9.42653
58 9.41205
47 9.42704
59 9.41252
48 9.42755
60 9.41 300
9.42805
1660
.
I
cas
1040
140
log cot
I d
2550 3460
I
10" cot
51
0.57703
51 0.57652
51[
0.576QI
51 0.57550
51 0.57499
51
0.57448
51 0.57 397
50 0.57347
51 0.57296
51 0.57 24~
50
0.57195
1
1
led!
log tan
3
3
3
3
3
4
3
3
3
3
3
4
3
3
3
3
4
3
3
3
4
3
3
3
4
3
3
3
4
3
3
3
I. 4
I 3.
I
4
3
3
3
4
3
3
I
j
3
4
3
3
4
3
60
59
58
57
56
66
54
53
52
51
60
49
48
47
46
46
44
43
42
41
40
39
38
37
36
36
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
16
14
13
12
11
9.98 52~ 3 10
9
9.98525
9.98 521
9. 98 51 ~
9. 98 515
9.98 511
9. 98 50~
9.98505
9.98 501
9.98 498
9.98 494
log sin
750
4
3
3
4
3
3
4
3
4
Id
8
7
6
6
4
3
2
1
0
1
2
3
4
5
6
7
8
9
10
20
30
40
50
64
0.9
1.8
2.7
3.6
4.5
5.4
6.3
7.2
8.1
9.0
18.0
27.0
36.0
45.0
I
61
1
0.8
2
1.7
3
2.6
4
3.4
54.24.24.1
65.15.04.9
7
6 0
8
6.8
9
7.6
8.5
10
20 17 0
30 I 25 5
40 34.0
50 42.5
I
I
1
2
3
4
5
6
7
~
10
20
30
40
50
48
0.8
1.6
2.4
3.2
4.0
4.8
5.6
6.4
7.2
8.0
16.0
24.0
32.0
40.0
15°
I log
TABLE III
1940 284°
,
Prop. Parts
d
63
0.9
1.8
2.6
3.5
4.4
5.3
6.2
7.1
8.0
8.8
17.7
26.5
35.3
44.2
62
0.9
I. 7
2.6
3.5
4.3
52
6.1
6.9
7.8
8.7
17.3
26.0
34.7
43.3
60
0.8
1.7
2.5
3.3
49
0.8
1.6
2.4
3.3
5. &
6.7
7.5
8.3
16. 7
25.0
33.3
41.7
. 5.7
6.5
7.4
8.2
16 3
24.'>
32.7
40.8
0
1
2
3
4
6
6
7
8
9
10
11
\2
13
14
16
16
17
18
19
20
21
22
23
24
26
26
27
28
29
80
Id I
9.41300
9.41347
9.41 394
9.41441
9.41 48~
9.41535
9.4\ 582
9.41 628
9.41675
9.41722
9.41 76~
9.41815
9.41861
9.41908
9.41954
9.42001
9.42047
9.42093
9.42140
9.42186
9.42232
9.42278
9.42324
9.42370
9.42416
9.42461
9.42 507
9.42553
9.42599
9.42 644
47
47
47
47
47
47
46
47
47
46
47
46
47
46
47
46
46
47
46
46
46
46
46
46
45
:~
46
45
46
log tan led I log cot
9.42805
9.42856
9.42906
9.42957
9.43007
9.43057
9.43 \08
9.43158
9.43208
9.43258
9.43308
9.43358
9.43408
9.43458
9.43508
9.43558
9.43607
9.43657
9.43707
9.43756
9.43806
9.43855
9.43905
9.43954
9.44004
9.44053
9.44102
9.44151
9.44201
9.44 250
0.57195
51 0.57144
50 0.57094
51 0.57043
50
0.56993
50
0.56943
51
SO 0.56892
50 0.56842
50 0.56792
50 0.56742
50 0.56692
50 0.56642
50 0.56592
50 0.56542
50 0.56492
49 0.56442
50 0.56393
50 0.56343
49 0.56293
50 0.56244
49 0.56191
50 0.56145
49 0.56095
50 0.56046
49 0.55996
49 0.55947
4910.55898
50 0.55849
49 0.55799
49 0.55 750
..
36
9.42917
36
37
38
39
40
41
42
43
44
46
46
47
48
49
60
51
9.42 962
9.43008
9.43053
9.43098
9.43143
9.43188
9.43233
9.43278
9.43323
9.43367
9.43412
9.43457
9.43 502
9.43546
9.43591
9.43635
52
53
54
66
56
57
58
59
60
9.43680
9.43724
9.43 769
9.43813
9.43857
9.43901
9.43946
9.43990
9.44 034
~5 '9.44544
46 9.44 592
45 9.44641
45 9.44690
45 9.44738
45 9.44787
45 9.44836
45 9.44884
45 9.44933
44 9.44981
45 9.45029
45 9.45078
45 9.45126
44 9.45 174
459.45222
44 9.45271
45 9.45319
44
45
44
44
44
45
44
44
Id
10" cas
164 0 2540 344: 0
9.45 36Z
9.45415
9.45 463
9.45511
9.45559
9.45606
9.45654
9.45702
9.45 750
10" cot
9.98494
9.98491
9.98488
9.98484
9.9848\
9.98477
9.98474
9.98471
9.98467
9.98464
9.98460
9.98457
9.98453
9.98450
9.98447
9.98443
9.98440
9.98436
9.98433
9.98429
9.98426
9.98422
9.98419
9.98415
9.98412
9.98409
9.98405
9.98402
9.98398
3
3
4
3
4
3
3
4
3
4
3
4
3
3
4
3
4
3
4
3
4
3
4
3
3
4
3
4
3
9.98 395 4
9.98391 3
9.98 388 4
9.98384
31
30
29
28
9.98
q q8 381
377 . ~
I 'A'"
I
0.554569.98373
~8''
27
1
I
1060
Id I I
cas
60
59
58
57
56
66
54
53
52
51
60
49
48
47
46
46
44
43
42
41
40
39
38
37
36
36
34
33
32
30 9.42690 45 9.44299 49,0.55701
31 9.42 735 46 9.44 348
0.55 652
32 9.42781
9.44397 49',0.55603
33 9.42 826
9.44
i :~' 0.55 554
Q 44 446
4o'i'
n 55 505
34 9 47 877, I :~
47 4
3
0.8 0.1 0.0
1.60.10.1
2.4 0.2 0.2
3.1 0.30.2
3.9 0.3 0.2
4.7 0.4 0.3
5.5 0.5 0.4
6.3 0.5 0.4
7.0 0.6 0.4
7.8 0.7 0.5
15.7 1.3 1.0
23.5 2.0 1.5
31.3 2.7 2.0
39.2 3.3 2.5
Prop. Parts
I
log sin
1
TABLE III
,
49 0.55 408
49 0.55359
48 0.55310
49 0.55262
49 0.55213
48 0.55164
49 0.55116
48 0.55067
48 0.55019
49 0.54971
48' 0.54922
48 0.54874
48 0.54 826
490.54778
48 0.54729
48 0.54681
1
1
48
48
48
48
47
48
48
48
led!
1
Prop. Parts
61
1
0.8
2
1.7
3
2.6
4
3.4
54.24.24.1
6
5.1
7
6.0
8
6.8
97.67.574
108.58.382
20 \7.0
30 25.5
40 34.0
50 42.5
60
0.8
1.7
2.5
3.3
49
0.8
1.6
2.4
3.3
5.0
5.8
6.7
4.9
5.7
6.5
16.7
25.0
33.3
41.7
16.3
24.5
32.7
40.8
47
0.8
1.6
2.4
3. I
3.9
4.7
5.5
6.3
7.0
7.8
46
0.8
1.5
2.3
3.1
3.8
4.6
5.4
6.1
6.9
7.7
1
-3
9.98 370
9.98366
9.98363
9.98359
9.98356
9.98352
9.98349
9.98345
9.98342
9.98338
9.98334
9.98331
9.98 327
9.98324
9.98320
9.98317
0.54633
9.98313
0.54585
9.98309
0.54 537 9.98 306
0.54489
9.98302
0.54441
9.98299
0.54394
9.98295
0.54346
9.98291
0.54298
9.98 288
0.54 250 9.98 284
102: tan
1950 2860
log sin
74°
26
4 24
3 23
4 22
3 21
4 20
3 19
4 18
3 17
4 16
4 16
3 14
4 13
3 12
4 11
3 10
[
4
4
3
4
3
4
4
3
4
d
9
8
7
6
6
4
}
2
1
0
1
2
3
4
5
6
7
8
9
10
20
30
~~
46
44 4
0.8 0.7 O. I
1.51.50.10.1
2.2 2.2 0.2
3.0 2.9 0.3
3.8 3.7 0.3
4.5 4.4 0.4
5.2 5.10.50.4
6.0 5.9 0.5
6.8 6.6 0.6
7.5 7.3 0.7
15.0 14.7 1.3
22.522.02.0
30.0 29.3 2.7
37.5 36.7 3.3
3
0.0
0.2
0.2
0.2
0.3
0.4
0.4
0.5
1.0
1.5
2.0
2.5
Prop. Parts
81
12
13
14
15
16
9.44559
9.44602
9.44646
9.44689
9.44733
P 9.44776
\8 9.44819
19 9.44862
20 9.44905
21 9.44 948
22 9.44 99~
23 9.45035
24 9.45077
25 9.45120
26 9.45 \63
27 9.45206
28 9.45 249
29 9.45 292
30 9.45334
31 9.45377
9.45 750
9.45 797
9.45845
9.45892
9.45 940
9.45987
9.46035
9.46082
9.46130
9.46 177
9.46224,
43
44
43
44
43
43
43
43
43
44
43
42
43
43
43
43
43
42
43
42
I 43
I 47
9.46319
9.46366
9.46413
9.46460
9.46507
9.46554
9.46601
9.46648
9.46694
9.46 741
9.46 788
9.46835
9.46881
9.46928
9.46975
9.47 021
9.47 068
9.47 114
9.47160
9.47207
9.47 253
947299
43 9.46271
47 0.54 250
48 0.54 203
47 0.54155
0.54108
~~ 0.54 060
48 0.54013
47 0.53965
4810.53918
0.53870
~~ 0.53 823
47 0.53776
9.98 284
9.98 281
9.98277
9.98273
3
4
4
9.98 270
~
4710.53681
47 0.53634
47 0.53587
47 0.53540
0.53 493
47' 0.53446
47
47 0.53399
46 0.53352
47 0.53306
0.53 259
~~ 0.53212
46 0.53165
47 0.53119
47 0.53072
0.53025
~~, 0.52979
46 0.52 932
46 0.52 886
47 0.52840
0.52793
~~I 0.52 747
i 052701
9.98240
9.98237
9.98233
9.98229
9.98226
9.98222
9.98 21~
9.98215
9.98211
9.98 207
9.98204
9.98200
9.98196
9.98192
9.98189
9.98185
9.98 181
9.98 177
998174
9.98170
9.98166 i
I
.
I
1
1
48,0.53729
1
'
1
9.98266
9.98262
9.98259
9.98255
9.98 251
9.98248
9.98244
I
60
59
58
57
56
1
4
55
54
53
52
51
50
3
4
4
3
4
4
3
4
4
3
4
4
4
3
4
4
4
3
4
4
4
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
4
3
4
4
32 9.45419
33 945462
99816213
34 9.45 504 ! 43 9 47 346 1~! 0.52 654 9 98 15~ I 4
49
27
35 9.45547
4610.52608
42 9.47392
36 9.45 589 43 9.47 438 46 0.52 562
37 9.45632
42 947484
10.52516
38 9.45674
42 9.47 530 ~~ 0.52 470
39 9.45716
42 9.47576
46 0.52424
40 9.45758
43 9.47622
4 0.52378
141 9.45801
42 9.47 668 4~1 0.52 332
42 9.45843
42 947714
46 0.52286
43 9.45885
42 9.47760
46 0.52240
44 9.45927
42 9.47806
46' 0.52194
45 9.45969
42 9.47852
45 0.52148
46 9.46 011 42 9.47 897 46 ' 0.52 103
479.4605242947943460.520579.98110413
48 9.46095
41 947989
46 0.52011
,49 9.46136
42 9.48035
4510.51965
50 9.46 178 42 9 48 080 461 O. 51 920
51 9.46220
42 948126
45' 0.51874
52 9.46 262 41 9.48 171 46 0.51 829
53 9.46302
42 948217
45 0.51783
54 9.46 345 41 9.48 262 45 0.51 738
55 9.46 386 42 9.48 307 46 0.51 693
56 9.46428
41 948353
45 0.51647
57 9.46 469 42 9.48 398 45 0.51 602
58 9.46511
41 948443
46 0.51557
59 9.46552
42 9.48489
45 0.51 511
9.98155
9.98 15\
9.98147
9.98144
9.98140
9.98136
9.98132
9.98129
9.98125
9.98121
9.98117
9.98 113
4
4
3
4
4
4
3
4
4
4
4
3
26
25
24
23
22
21
20
19
18
17
16
15
14
9.98106
9.98102
9.98 098
9.98094
9.98 090
9.98087
9.98 083
9.98 079
9.98075
9.98 071
9.98067
9.98063
4
4
4
4
3
4
4
4
4
4
4
3
12
1\
10
9
8
7
6
5
4
3
2
1
60
9.98060
I
1
1
[
9.46594
\)
1630 2530 3430
9.48534
0.51
466
I
TABLE III
17°
logsin I d I logtan led I logcot I logcos I d
0 9.46594 41 9.48534 4510.51466 9.98060 4
1 9.46635
41 9.48579 45 0.51421 9.98056 4
Prop. Parts
48
47
1
0.8
0.8
21.61.61.5
3
2.4
2.4
4
3.2
3.1
5
4.0
3.9
6
4.8
4.7
7
5.6
5.5
8
6.46.36.1
9
7.2
7.0
10
8.0
7.8
20 16.0 15.7
30 24.0 23.5
40 32. 0 31. 3
50 40.0 39.2
45
1
0.8
2
1.5
3
2.2
4
3.0
5
3.8
6
4.5
7
5.2
8
6.0
9,
6.8
10 I 7 5
20 I 15 0
30 22.5
40 30.0
50 37.5
1
1
2
3
4
5
6
7
8
9
10
20
3~
~O
2
3
4
5
6
7
8
9
10
\1
12
13
14
15
16
17
18
19
20
2\
22
23
24
25
26
27
28
29
30
46
0.8
2.3
3.1
3.8
4.6
5.4
6.9
7.7
15.3
23.0
30. 7
38.3
44
0.7
1.5
2.2
2.9
3.7
4.4
5.'
5 i
6.6
7.3
14.7
22 . 0
29.3
36.7
42
41 4
0.7 0.70.1
1.41.41.10.1
2.1 2.00.20.2
2.8 2.7 0.3
3.5 3.4 0.3
4.2 4.1 0.4
4.9 4.8 0.5
5.6 5.5 0.5
6.3 6.2 0.6
7.0 6.8 0.7
14.0 13.7 1.3
21.020.52.0
28.0 27.3 2.7
35.0 34.2 3.3
43
0.7
1.4
2.2
2.9
3.6
4.3
5.0
5.7
6.4
7.2
14.3
21. 5
28.7
35.8
--
0.2
0.2
0.3
0.4
0.4
0.4
0.5
1.0
1.5
2.0
2.5
9.46758
41 9.48 759
41 9.48 804
41 9.48 849
41 9.48894
41 9.48939
40 9.48 984
41 9.49029
41 9.49073
41 9. 49 118
41 9.49163
409.49207
41 9.49252
40 9.49296
41 9.49341
40 9.49385
41 9.49430
40 9.49474
41 9.49519
40 9.49 563
40 9.49 607
41 9.49652
40 9.49 696
40 9.49740
40 9.49784
40 9.49 828
40 9.49872
0.51 241
45
45 ' 0.51 196
45 0.51 15\
45,0.51 106
45 0.51061
45 0.51 0' 6
44,0.50971
4510.50927
O.50 882
45'
4410.50837
450.50793
44,0.50748
4510.50704
0.50652
44' 0.50615
45
44' 0.50 570
45 0.50526
44 0.50481
44 0.50 437
45 0.50 393
4410.50348
44 0.50 304
4410.50260
4410.50216
0.50 172
441
44 0.50128
42 9.48714
9.46 800
9.46 841
9.46 882
9.46923
946 96~
9.47 005
9.47045
9.47086
9. 47 127
9.47168
9.47209
9.47249
9.47290
9.47330
9.47371
9.47411
9.47452
9.47492
9.47 533
9.47 573
9.47613
9.47 654
9.47694
9.47734
9.47 774
9.47814
4
4
45] 0.51286
9.98044
4
9.98 040
9.98 036
9.98 032
9.98022
9.98025
9.98 021
9.98017
9.98013
9. 98 009
9.98005
9.98001
9.97997
9.97993
9.97989
9.97986
4
4
3
4
4
4
4
4
4
4
4
4
4
3
4
9.97982
4
9.97978
9.97974
9.97 970
9.97 966
9.97962
9.97 958
9.97 9~4
9.97950
9.97946
9.97942
4
4
4
4
4
4
4
4
4
4
1
9.47 974
9.48014
9.48 054
9.48094
9.48 133
9.48 173
9.48 213
9.48252
9.48 292
9.48332
9.48371
9.48411
9.48 450
9.48490
9.48 529
9.48 568
40
40
40
39
40
40
39
40
40
39
40
39
40
39
39
39
9.50 048
9.50092
9.50 136
9.50 180
9.50 223
9.50 267
9. 50 311
9.50355
9.50 398
9.50442
9.50485
9.50529
9.50 572
9.50616
9.50 659
9.50 703
44' 0.49 952
4410.49908
44 0.49 864
43 0.49 820
44, 0.49 777
44 0.49 733
44 0.49 689
43 0.49645
44 0.49 602
9.97 926
9.97922
9.97 918
9.97914
9.97910
9.97 906
9.97 902
9.97898
9.97 894
I4
43 0.49 55~ 9.97890
0.49515
0.49471
0.49 428
0.49384
0.49 341
0.49 297
9.97886
9.97882
9.97 878
9.97874
9.97 870
9.97 866
4
1
1
4
4
4
4
4
4
4
4
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
11
10
9.48686
9.48725
9.48764
9.48 803
9.48842
9.48 881
9.48 920
9.48 959
9.48 998
39
39
39
39
39
39
39
39
39
Id
1620 2520 342~
9.50 789
9.50833
9.50876
9.509\9
9.50 962
9.51005
9.51 048
9.51 092
9.51 135
9.51 178
log cot
44
43
43
43
43
43
44
43
43
cd
0.49 211
0.49167
0.49124
0.49081
0.49 038
0.48995
0.48 952
0.48 908
0.48 865
0.48 822
9.97 857
9.97853
9.97849
9.97845
9.97 841
9.97837
9.97 833
9.97 829
9.97 825
9.97 821
4
4
4
4
4
4
4
4
4
log t:ia
log sin
d
72°
45
44
43
1
0.8
0.7
0.7
2
1.5
1.5
1.4
3
2.2
2.2
2.2
4
3.0
2.9
2.9
5
3.8
3.7
3.6
6
4.54.44.3
75.25.15.0
8
6.0
5.9
5.7
9
6.8
6.6
6.4
10
7.5
7.3
7.2
20 15.0 14.7 14.3
30 22.5 22.0 21.5
40 30.0 29. 3 28.7
50 37.5 36.7 35.8
40
5
f!O
9.48 607 40 9.50 746 43, 0.49 254 9.97 861 4
)
I 9.48 647
52
53
54
55
56
57
58
59
60
...
59
58
57
4
4
4
4
4
44
43'
44
43
44
43
1
1
0
60
16
15
14
13
12
.og cos
73°
0.51 376 9.98 052
45
4510.51331
9.98048
1
og co
82
41 9.48 624
41 9.48669
31 9.47 854 40 9.49916 I 44 0.50 084 9.97 938 ! 4
32 9.47 894 40 9.49 960 1 44: 0.50 040 9.97 934 i 4
33 9.47934
9.97930
44 0.499%
4
I 40 9.50004,
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
3
0.0
9.46 676
9.46717
1070 1970 2870
Prop. Parts
9
8
7
6
5
4
3
2
1
0
1
2
3
4
5
6
7
8
42
0.7
1.4
2.1
2.8
3.5
4.2
4.9
I
5.6
1
11 9.44516
44
44
44
4
41
44
44
44
43
44
44
log cot
1
9.44 034
9.44 078
9.44122
9.44166
9.44 210
9.44253
9.44297
9.44341
9.44385
9.44 428
9.44472
1060 1960 2860
Id I
I
1
0
I
2
3
4
5
6
7
8
9
10
16°
log cos
cd
1
1
TABLE III
log sin I d , log tan
9
10
20
30
40
50
6.3
7.0
14.0
21.0
28.0
35.0
41
0.7
1.4
2.0
2.7
3.4
4.1
4.8
5.5
6.2
6.8
13.7
20.5
27.3
34.2
40
0.7
1.3
2.0
2.7
3.3
4.0
4.7
5.3
6.0
6.7
13.3
20.0
26.7
33.3
39543
10.60.10.10.0
21.30.20.10.1
3
2.00.2
0.2 0.2
4
2.6 0.3 0.30.2
53.2040.30.2
6
3.90.5
0.4 0.3
7
4.60.60.50.4
8
5.20.70.50.4
9
5.80.80.60.4
10
6.5 0.8 0.7 0.5
20 13.0 1. 7 1. 3 1. 0
30 19. 5 2. 5 2. 0 I. 5
40 26.0 3.3 2.7 2.0
50 ' 32.5 4.2 3.3 2.5
I
Prop.
Parts
83
180
TABLE III
log sin
d
I
log tan
I cd
9.48 998 39 9.51 178
9.51221
9.49037
39 9.51 264
9.4907§
39
9.49 115 38 9.51 306
9.51349
9.49153
39
9.49 192 39 9.51 39£
9.49231
38 9.51435
9.49269
39 9.51478
9.49308
39 9.51 520
9.49347
38 9.51563
9.49 385 39 9.5\ 606
9.49424
38 9.51 648
9.49 462 38 9.51 691
9.49500
39 9.51734
9.49539
38 9.51776
9.49577
38 9.51819
9.49 615 39 9.51 861
9.49 654 38 9.51 903
9.49692
38 9.51946
9.49730
38 9.51988
9.49768
38 9.52 031
9.49806
38 9.52 073
9.49844
38 9.52 115
9. 49 882 38 9.52157
9.49920
38 9.52200
9.49958
38 9.52242
9.49996
38 9.52 284
9.50034
38 9.52 326
9.50072
38 9. 52 368
9.50110
38 9.52 410
9.50148
37 9.52452
9.50185
38 9.52494
9.50223
389.52536
9 50261.
37 9.52578.
9.50298!
38 9.52620
9.50336
38 9.52 661
9.50374
37 9.52703
9.50411
38 9.52 745
9.50449
37 9.52787
9.50486
37 9.52829
9.50 523 38 9.52 870
41 9.50561
37 9.52912
42 9 . 50 598 37 9.52953
43 9.50635
38 9.52 995
44 9.50673
37 9.53037
45 9.50710
37 9.53078
46 9.50747
37 9.53120
47 9.50 784 37 9.53161
48 9.50821
37 9.53202
49 9.50858
38 9.53244
50 9.50896
37 9.53285
51 9.50933
37 9.53327
52 9.50 970 37 9.53368
53 9.51 007 36 9.53409
54 9.51043
37 9.53450
55 9 . 51 080 37 9.53492
56 9.51 117 37 9.53533
57 9.51 154 37 9.53574
58 9.51 \91 36 9.53615
59 9.51 227 37 9.53656
9.53697
60 9. 51 264
0
\
2
3
4
5
6
7
8
9
10
1\
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
I:
°
2510
3410
d
an
Prop. Parts
I
60
43 0.48 822 9.97 821 4 59
9.97817
5 58
43 0.48779
9.97812
4
42 0.48736
0.48 694 9.97 808 4 57
43
56
0.48651
9.97804
4
43
66
0.48
608
9.97
800
4
43
9.97796
4 54
43 0.48565
0.48522
9.97792
4 53
42
0.48480
9.97788
4 52
43
0.48437
9.97784
5 5\
43
0.48
394
9.97
779
4 50
42
9.97775
4 49
43 0.48352
48
43 0.48 309 9.97 771 4 47
9.97767
4
42 0.48266
9.97763
4 46
43 0.48224
0.48181
9.97759
5 45
42
0.48
139
9.97
754
4 44
42
0.48
097
9.97
750
4 43
43
0.48054
9.97746
4 42
42
0.48012
9.97742
4 41
43
420.47969
9.97738
4 40
5 39
42 0.47 92Z 9.97734
420.47885
9.97729
4 38
430.47843
9.97725
4 37
420.47800
9.97721
4 36
420.47758
9.97717
4 36
9.97713
5 34
42 0.47716
420.47674
9.97708
4 33
9.97704
4 32
42 0.47632
420.47590
9.97700
4 31
30
0.47548
9.9769615
42
9.97691
4 29
42 0.47506
42,0.474649.97687
4 28
4210.47422
9.97683
4 27
I 4110.47380
9.97679!
5 26
420.47339
9.97674
4 25
9.97670
4 24
42 0.4729Z
9.97666
4 23
42 0.47255
420.47213
9.97662
5 22
9.97657
4 21
41 0.4717\
0.47130
9.97653
4 20
42
9.97642
4 19
41 0.47088
420.47047
9.97645
5 18
9.97640
4 17
42 0.47005
16
41 0.46 963 9.97 636 4
9.97632
4 15
42 0.46922
14
4\ 0.46 880 9.97 628 5 13
41 0.46 839 9.97 623 4
420.46798
9.97612
4 12
9.97615
5 11
41 0.46756
9.97610
42 0.46715
4 10
9
9.97606
41 0.46673
4
8
9.97602
41 0.46632
5
7
9.97597
41 0.46521
4
6
9.97593
42 0.46550
4
5
9.97589
41 0.46508
5
4
9.97584
41 0.46467
4
3
9.97580
41 0.4642§
4
2
9.97576
4\ 0.46385
5
1
9.97571
41 0.46344
4
0
0.46 303 9.97 567
0
161
log cos
log cot
43
0.7
1.4
2.2
2.9
3.6
4.3
5.0
5.7
6.4
7.2
14.3
21.5
28.7
35.8
\
2
3
4
5
6
7
8
9
10
20
30
40
50
1
2
3
4
5
6
7
8
9
10
20
30
40
50
39
0.6
1.3
2.0
2.6
3.2
3.9
4.6
5.2
5.8
i 6.5
i 13. 0
5
119.
26.0
32.5
1
2
3
4
5
6
7
8
9
10
20
30
40
50
36
0.6
1.2
1.8
2.4
3.0
3.6
4.2
4.8
5.4
6.0
12.0
18.0
24.0
30.0
41
0.7
1.4
2.0
2.7
3.4
4.1
4.8
5.5
6.2
6.8
13.7
20 . 5
27.3
34.2
38
0.6
1.3
1.9
2.5
3.2
3.8
4.4
5.1
5.7
6.3
12.7
19 . 0
25.3
31.7
37
0.6
1.2
1.8
2.5
3.1
3.7
4.3
4.9
5.6
6.2
12;
18.5
24.7
30.8
5
0.1
0.2
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.8
1.7
2.5
3.3
4.2
4
0.1
0.1
0.2
0.3
0.3
0.4
0.5
0.5
0.6
0.7
1.3
2.0
2.7
3.3
0
1
2
3
4
6
6
7
8
9
10
11
12
13
14
16
16
17
\8
19
20
21
22
23
24
26
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
46
46
47
48
49
50
51
52
53
54
65
56
57
58
59
60
84
log tan
d
cd
log
~ot
I
9.51 264 37 9.53 697 41 0.46 303
0.46262
9.53738
9.51301
41 0.46 221
9.51 338 37
36 9.53 779 41 0.46180
9.51374
4\ 0.46139
37 9.53820
9.53861
9.51411
41
36
0.46098
9.51447
41 0.46057
37 9.53902
9.51484
41 0.46016
36 9.53943
9.51520
4\
37 9.53984
9.51557
40 0.45 97~
36 9.54025
9.51 593 36 9.54 065 41 0.45 935
9.51629
41 0.45894
37 9.54106
9.51 666 36 9.54 147 40 0.45853
9.51 702
9.54 187
0.45 813
9.51 738
~~9.54228 :1 0.45772
9.51774
40 0.45731
37 9.54269
9.51811
41 0.45691
36 9.54309
9.51 847 36 9.54 350 40 0.45 650
0.45610
9.51883
36 9.54390
0.45569
9.51919
36 9.54 431 :~
9.51 955 36 9.54471
41 0.45529
9.51991
4 0.45488
36 9.54512
9.52 027 36 9.54 552 4 ~0.45 448
9.52 063 36 9.54 593 40 0.45 407
9.52099
4 0.45367
36 9.54633
9.52 135 36 9.54 673 4~ 0.45327
9.52171
40 0.45286
36 9.54714
9.52 207 35 9.54 754 40 0.45 246
9.52242
41 0.45206
36 9.54794
9.52 278 36 9.54 835 40 0.45 165
9.52314
40 0.45125
36 9.54875
9.54915
9.52350
40 0.45 08~
35
9.52 385 36 9.54 955 40 0.45 045
9.52421
40 0.45002
35 9.54995
9.52 456 I 36 9.55 035 I 401 O. 44 96~
9.52 492 35 9.55 075 . 40 0.44 925
i
1
9.52 527 36 9.55 115 40' 0.44 885
9.55
155
9.52 563 35
40 0.44 845
9.52 598 36 9.55 195 40 0.44 805
40 0.44765
9.52634
35 9.55235
40 0.44725
9.52669
36 9.55275
9.55312
40 0.44685
9.52705
35
40 0.44645
9.52740
35 9.55355
39 0.44605
9.52775
36 9.55395
40 0.44566
9.52811
35 9.55434
4010.44526
9.52846
35 9.55474
40 0.44486
9.52881
35 9.55514
9.52916
39 0.44446
35 9.55554
40 0.44407
9.52951
35 9.55593
9.52 986 35 9.55 633 40 0.44 367
9.53 021 35 9.55 673 39 0.44 327
9.53056
40 0.44 288
36 9.55712
9.53092
34 9.55 752 39 0.44 248
9.53126
40 0.44 209
35 9.55791
9.53 161 35 9.55 831 39 0.44169
9.53196
35 9.55 870 40 0.44 130
9.53231
359.55910
390.44090
40 0.44051
9. 53 266 35 9.55949
9.53301
35 9.55 989 39 0.44 011
9.53 336 34 9.56 028 39 0.43 972
9.53370
35 9.56 067 40 0.43 933
0.43 893
9.56107
9.53405
10
0
71°
log sin
'
42
0.7
1.4
2.1
2.8
3.5
4.2
4.9
5.6
6.3
7.0
14.0
21. 0
28.0
35.0
190
TABLE III
1080 1980 288°
cos
d
~160°250° 3400
I
log
cot
Ic d I
log
tan
109°
9.97 567 4
9.97563
5
9.97 558 4
9.97 5~4 4
9.97550
5
9.97545
4
9.97541
5
9.97536
4
9.97532
4
9.97 528 5
9.97523
4
9.97519
4
9.97 515
9.97 510
9.97506
5
9.97501
4
9.97 497 5
9.97492
9.97488
9.97484
5
9.97479
4
9.97 475 5
9.97 470 4
9.97466
9.97 461
9.97457
4
9.97 453 5
9.97448
4
9.97 444 5
9.97439
4
9.97435
5
9.97 430 4
9.97426
5
9.97 421 4
9 97 417 . 5
1
9.97 412 4
9.97 408 5
9.97 403 4
9.97399
5
9.97394
4
9.97390
5
9.97385
4
9.97381
5
9.97376
4
9.97372
5
9.97367
4
9.97363
5
9.97358
5
9.97 353 4
9.97 349 5
9.97 344 4
9.97 340 5
9.97 335 4
9.97331
5
9.97 326 4
9.97322
5
9.97317
5
9.97 312 4
9.97 308 5
9.97 303 4
9.97 299
:
I
log
sin
1990 2890
Prop. Parts
log cos
I
d
60
59
58
57
56
66
54
53
52
51
60
49
48
~47
46
46
44
43
42
41
40
39
38
37
~36
36
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
16
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
I
'
41
0.7
1.4
2.0
2.7
3.4
4.1
4.8
5.5
6.2
6.8
13.7
20.5
27.3
34.2
\
2
3
4
5
6
7
8
9
10
20
30
40
50
37
0.6
1
1.2
2
1.8
3
2.5
4
3.1
5
3.7
6
4.3
7
4.9
8
5.6
9
6.2
10
20 1 12.3
j 18.5
30
40 24.7
50 30.8
I
1
2
3
4
5
6
7
8
9
10
20
30
40
50
I
40
0.7
1.3
2.0
2.7
3.3
4.0
4.7
5.3
6.0
6.7
13.3
20.0
26.7
33.3
39
0.6
1.3
2.0
2.6
3.2
3.9
4.6
5.2
5.8
6.5
13.0
19.5
26.0
32.5
36
0.6
1.2
1.8
2.4
3.0
3.6
4.2
4.8
5.4
6.0
12 0
\8.0
24.0
30.0
35
0.6
1.2
1.8
2.3
2.9
3.5
4.1
4.7
5.2
5.8
11 7
17.5
23.3
29.2
34
0.6
1.1
1.7
2.3
2.8
3.4
4.0
4.5
5.1
5.7
11.3
17.0
22.7
28.3
5
0.1
0.2
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.8
1.7
2.5
3.3
4.2
Prop.
Parts
4
0.1
0.1
0.2
0.3
0.3
0.4
0.5
0.5
0.6
0.7
1.3
2.0
2.7
3.3
d
9.53405
9.53 44Q
9.53475
9.53509
9.53 544
9.53578
9.53 613
9.53647
9.53682
9.537\6
9.53751
9.53785
9.53819
9.53854
9.53 888
9.53922
9.53957
9.5399!
9.54025
9.54 059
9.54 093
9.54127
9.54161
9.54 195
9.54229
9.54263
9.54 297
9.5433!
9.54365
9.54 399
9.54433
9.54 466
9.54500.
9.54534
9.54567.
9.54 60!!
9.54635
9.54668
9.54 702
9.54735
9.54769
9.54 802
9.54836
9.54869
9.54903
9.54936
9.54969
9.55 003
9.55036
9.55069
9.55102
9.55 136
9.55 169
9.55 20~
9.55235
9.55268
9.55301
9.55 334
9.55 367
9.55 400
9.55433
led
I
0.43893
9.56107
35 9.56 146 39 0.43 851
39 0.43815
35 9.56185
39
34
359.56224
400.43776
0.43 736
9.56
264
39
34
0.43697
9.56303
39 0.43 658
35
34 9.56 342 39 0.43619
9.56381
39 0.43580
35
39
34 9.56420
0.43541
9.56459
39
35
0.43502
9.56498
39
34
39 0.43463
34 9.56537
39 0.43421
35 9.56576
39 0.43385
34 9.56615
0.43 346
34 9.56 654 39
39 0.43307
35 9.56693
39 0.43268
34 9.56732
39 0.43229
34 9.56771
39 0.43190
34 9.56810
0.43 15\
34 9.56 849 38
9.56
887
39 0.43 113
34
39 0.43074
34 9.56922
39 0.43035
34 9.56965
996
34 9.57 004 38 0.42
39 0.42958
34 9.57042
39 0.42919
34 9.57081
880
34 9.57 120 38 0.42
39 0.42842
34 9.57158
38 0.42 80~
34 9.57197
34 9.57 235 39 0.42 765
38 0.42726
33 9.57274
34 9.57 312 39 0.42 688
38 0.42649
34 9.57351
39 0_4261\
33 9.573891
34 9.57428.30.0.42572
38' 0.42534
34 9.57466'
39 0.42496
33 9.57504
38 0.42457
34 9.57543
33 9.57 581 38 0.42 419
39 0.42381
34 9.57619
9.57658
38 0.42342
33
34 9.57 696 38 0.42 304
38 0.42266
33 9.57734
38 0.42228
34 9.57772
39 0.42190
33 9.57810
38 0.42151
33 9.57849
38 0.42113
34 9.5788£
33 9.57 925 38 0.42 075
38 0.42037
33 9.57963
38 0.41999
33 9.58001
34 9.58039
38 0.41961
33 9.58 01£ 38 0.41 923
33 9.58 115 38 0.41 885
33 9.58153
38 0.41847
33 9.58191
38 0.41809
38 0.41771
33 9.58229
37 0.41733
33 9.58267
33 9.58 304 38 0.41 696
33 9.58 342 38 0.41 658
33 9.58 380 38 0.41 620
9.58418
0.41582
i
1100 2000 290°
log cos
log cot
9.97 299
9.97 294
9. 97 28~
9.97 285
9.97 280
9.97 276
9.97 271
9.97266
9.97 262
9.97 257
9.97 252
9.97 248
9.97 243
9.97 238
9.97 234
J.97 229
9.97 224
9.97 220
9.97215
9.97 2\ 0
9.97 206
9.97 20I
9.97 196
9.97192
9.97 187
9.97 182
9.97 178
9.97173
9.97 168
9.97 163
9.97 159
9.97 154
9. 97 14~
9.97145
5
5
4
5
4
5
5
4
5
5
4
5
5
4
5
5
4
5
5
4
5
5
4
5
5
4
5
5
5
4
5
5
4
5
9 97 140 5
9.97135' 5
9.97 130 4
9.97 126 5
9.97121 5
9.97 116 5
9.97 111 4
9.97107 5
9.97 102 5
9.97 097 5
9.97092 5
9.97 087 4
9.97 083 5
9.97 078 5
9.97 073 5
9.97 068 5
9.97 063 4
9.97 059 5
9.97 054 5
9.97 049 5
9.97 044 5
9. 97 03~ 4
9.97 035 5
9.97 03Q 5
9.97025 5
9.97 020 5
9.97 015
.
1
1
n
1690
2490
3390
'6
d
60
59
58
57
56
66
54
53
52
51
60
49
48
47
46
46
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
1
2
3
4
5
6
7
8
9
10
20
30
40
50
log sin
40
39
0.7
0.6
1.3
1.3
2.0
2.0
2.7
2.6
3.3
3.2
4.0
3.9
4.7
4.6
5.3
5.2
6.0
5.8
6.7
6.5
13. 3 13.0
20.0 19.5
26. 7 26.0
33.3 32.5
38
0.6
1.3
1.9
2.5
3.2
3.8
4.4
5.1
5.7
6.3
12.7
19.0
25.3
31.7
37
0.6
1.2
1.8
2.5
3.1
3.7
4.3
4.9
5.6
6.2
12 3
34
0.6
1. 1
1.7
2.3
2.8
3.4
4.0
4.5
5.1
5.7
11:-J
17.0
22 . 7
28.3
0
1
2
3
4
5
6
7
8
9
10
11
12
13
\4
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
.
35
0.6
1
1.2
2
1. 8
3
2.3
4
5
'.9
j.5
6
4.1
7
4.7
8
5.2
9
5.8
10
1\ 7
20
30 i 18.5 17 5
40 24. 7 23 . 3
50 30. 8 29.2
I
1
2
3
4
5
6
7
8
9
10
20
30
40
50
33
5
0.6 0.1
1.10.20.1
1.60.20.2
2.2 0.3
2.8 0.4
3.3 0.5
3.8 0.6
4.4 0.7
5.0 0.8
5.5 0.8
11. 0 1.7
16. 5 2 . 5
22.0 3.3
27.5 4.2
4
0.1
0.3
0.3
0.4
0.5
0.5
0.6
0.7
1.3
2.0
2.7
3.3
2r
TABLE III
Prop. Parts
d
Id
9.55433
9.55466
9.55 499
9.55 532
9.55564
9.55 597
9.55630
9.55 663
9.55 695
9.55728
9.55761
9.55 793
9.55 826
9.55858
9.55 891
9.55 923
9.55 956
9.55 988
9.56 021
9.56 053
9.56085
9.56118
9.56150
9. 56 18~
9.56 2\5
9.56247
9.56 279
9.56311
9.56343
9.56375
9.56 408
9.56440
9.56472
9.56504
9 56 536
9.56568!
9.56599
9.56 631
9.56 663
9.56 695
9.56727
9.56 759
9.56 790
9.56822
9.56 854 I
9.56 886
9.55917
9.56 949
9 56 980
9.57 0 \2
9.57044
9.57 075
9.57107
9.57138
9.57169
9.57201
9.57 232
9.57261
9.57295
9.57 326
9.57 358
.
Prop. Parts
1580
log tan
33 9.58418
33 9.58455
33 9.58 493
32 9.58 531
33 9.58569
33 9.58 606
33 9.58644
32 9.58 681
33 9.58 719
33 9.58757
32 9.58794
33 9.58 832
32 9.58 869
33 9.58907
32 9.58 944
33 9.58 981
32 9.59 019
33 9.59 056
32 9.59 094
32 9.59 131
33 9.59168
32 9.59205
32 9.59243
33 9.59 280
32 9.59 317
32 9.59354
32 9.59 391
32 9.59429
32 9.59466
33 9.59503
32 9.59 540
32 9.59577
32 9.59614
3 9.59651
32 9 59 688;
31 9.59725'
32 9.59762
32 9.59 799
32 9.59 835
32 9.59 872
32 9.59909
31 9.59 946
32 9.59 983
32 9.60019
32 9.60 056
31 9.60 093
32 9.60130
31 9.60 166
32 9.60 203
32 9.60 240
319.60276
32 9.60 3\3
31 9.60349
31 9.60386
32 9.60422
31 9.60459
32 9.60 495
31 9.60532
31 9.60568
32 9.60 605
9.60 641
2480 3380
I log
cd
cot
0.41 58~
371 0.41545
38
381 0.41 507
381 0.41 469
0.41431
37 '
38 0.41 394
3710.41356
38 0.41 319
38 0.41 281
3710.41243
38 0.41206
37 0.41 168
38 0.41 \31
37 0.41093
37 0.4\ 056
38 0.41 019
37 ' 0.40 981
38 0.40 944
37 0.40 906
0.40 869
371
37 0.40 83~
38 0.40795
37 0.40757
37 0.40 720
37 0.40 683
37 0.40646
38 0.40 609
37 0.40571
37 0.40534
37 0.40497
37 0.40 460
37 0.40423
37 0.40386
3710.40349
37: 0 40 312
37 ' 0.40275
37,0.40238
361 0.40 20!
37 0.40 165
37 0.40 128
37 0.40091
37 0.40 054
36 0.40 017
37 0.39981
37 0.39 944
37 0.39 907
36 0.39870
37 0.39834
37 0.39 797
36 0.39 760
370.39724
36 0.39 687
37 0.39651
36 0.39614
37 0.39578
36 0.39541
37 0.39 505
36 0.39468
37 0.39432
36 0.39 395
0.39 359
1
1
log sin
0
1
2
3
4
6
6
7
8
9
10
11
12
13
14
16
16
17
18
19
20
21
22
23
24
26
26
27
28
29
30
31
32
33
34
36
36
37
38
39
40
41
42
43
44
46
46
47
48
49
60
51
52
53
54
66
56
57
58
59
60
log tan
1
20°
TABLE III
[
1
r
111 0 2010 29P
Id I
log cos
9.97015
9.97010559 5
9.97 005
4
9.97001
9.96 996 5
5
9. 96 991
9.96 986 5
9. 96 981 5
5
9.96 976 5
9.96 971
5
9.96 966 4
9.96 962 5
9.96 957 5
9.96952
5
9. 96 947 5
9.96942
5
9.96937
5
9.96932
5
9.96 927 5
9.96 922 5
9.96 917 5
9.96 912 5
9. 96 907 4
9.96 903 5
9.96 898 5
9.96893
5
9. 96 888 5
9.96883
5
9 96 878 5
9.96873
5
9. 96 868 5
9.96 863 5
9.96 858 5
9.90 853
i5
9 9(; 848, 5
9. 96 843 1 5
9.96 838 5
9.96 833 5
9.96 828 5
9.96 823 5
9. 96 818 5
9. 96 813 5
9. 96 808 5
9. 96 803 5
9.96 798 5
9.96 793 5
9.96788
5
9.96783
5
9.96778
6
9.96 772 5
9.96 767! 5
9.96762
5
9.96757
5
9.96 752 5
9.96747
5
9.96742
5
9.96 737 ' 5
9.96 732 5
9.96 727 I 5
9. 96 722 ,5
9 . 96 717
I
8~
1
I
I
Prop. Parts
60
58
57
56
55
54
53
52
5\
50
49
48
47
46
45
44
43
42
4\
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
38
1
0.6
21.31.2
3
1.9
4
2.5
53.23.13.0
6
3.8
7
4.4
85.14.94.8
9
5.7
10
6.3
20 12. 7
30 19. 0
40 25.3
50 3 I. 7
33
1
0.6
2
1.1
3
1.6
42.22.12.1
5
2.8
63.33.23.1
7
3.8
84.44.34.1
9,
5.0
10 i 5.5
37
0.6
1.8
2.5
36
0.6
1.2
1.8
2.4
3.7
4.3
3.6
4.2
5.6
5.4
6.2
6.0
12. 3 12. 0
18. 5 18. 0
24. 7 24.0
30. 8 30. 0
32
0.5
1.1
1.6
31
0.5
1.0
I. 6
2.7
2.6
3.7
3.6
4.8
5.3
4.6
5.2
20 11.0 10.7 10.3
30 ' 16 5 16.0 15.5
40 22. 0 21.3 20.7
.
50
I
27.5
26. 7 25.8
6
5
1 0.1 0.1
20.20.20.1
3 0.3 0.2
4 0.4 0.3
5 0.5 0.4
6 0.6 0.5
7 0.7 0.6
8 0.8 0.7
9 0.9 0.8
10 1.0 0.8
20 2 . 0 I. 7
30 3 . 0 2 . 5
40 4 . 0 3 . 3
50 5 .0 4 . 2
4
0.1
0.2
0.3
0.3
0.4
OS
0.5
0.6
0.7
I. 3
2 .0
2 .7
3. 3
137
l
TABLE III
log sin
0
I
2
3
4
5
6
7
8
9
10
II
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
9.57 358
9.57389
9.57420
9.57451
9. 57 482
9.57514
9.57545
9.57 576
9.57607
9. 57 638
9. 57 669
9.57700
9.57731
9.57762
9.57793
9. 57 824
9.57855
9. 57 885
9.57916
9.57947
9.57 978
9. 58 008
9.58039
9.58070
9.58 101
9.58 131
9. 58 162
9. 58 192
9.58 223
9.58253
9.58 284
9.58 311
9.58 345
9.58 375 i
I
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
9 58 406
I
9.58436
9. 58 467
9.58 497
9.58 527
9.58557
9. 58 588
9.58618
9.58 648
9.58 678
9.58709
9.58739
9. 58 769
9. 58 799
9.58 829
9.58859
9. 58 889
9.58919
52
53
54
55
56
57
58
59
60
9.58 949
9.58 979
9.59009
9.59 039
9.59 069
9.59098
9.59128
9. 59 158
9.59 188
log tan
d
.
Ic d I
1120 20202920
22°
log cot
log cos
31
31
31
32
31
31
31
31
3\
31
31
31
31
31
31
30
31
31
31
30
31
31
31
30
31
30
31
30
31
30
31
30
31
3610.393599.96717
9.60 677 370.39323
9.96711
9.60714
0.39 286 9.96 706
36
9. 60 750 36 0.39 25:> 9.96 701
9.60786
37 O.39 214 9. 96 696
9.60 823 36 0.39 177 9.96 691
9. 60 859
0.39 141 9.96 686
9.60 895 361
36,0.391059.96681
9. 60 931 361 0.39 069 9.96 676
9.60967
37 0.39 033 9.96 670
9. 61 004 36 0.38 996 9.96 665
9. 61 040 36 O.38 960 9. 96 660
9.61076
36 0.38 924 9. 96 6~5
9.61 112 36 0.38 888 9.96 65Q
9. 61 148 36 0.38 852 9.96 645
9.61 184 36 O.38 816 9. 96 640
9. 61 220 36 O.38 780 9. 96 634
9.61 256 36 0.38 744 9.96 629
9.61 292 36 0.38 708 9.96 624
9.61 328 36 0.38 672 9.96 619
9. 61 364 36 0 38 636 9.96 614
9.61400
36 O.38 600 9.96 608
9.61 436 36 O.38 564 9.96 603
9.61472
9.96598
36 0.38528
9. 61 508 36 0.38 492 9.96 593
9.96588
9. 61 544 35 0.38456
9.96582
9. 61 579 36 0.38421
9.61615
9.96577
36 0.38385
9. 61 651 36 0.38 349 9.96 572
9.61 687 35 O.38 313 9. 96 567
9.61 722 36 0.38 278 9.96 562
9.61 758 36 0.38 242 9.96 556
9.61 794 36 0.38 206 9.96 551
9.61 830 i 35i 0.38 17Q 9.96 546
9.61 865 I
38 135 9 96 541 I
31
30
30
30
31
30
9.61901
9.61 936
9.61 972
9.62 008
9.62043
9.62 079
31
9. 60 64\
30 9. 62 114
30 9. 62 I50
31 9.62 185
30 9.62 221
30 9.62256
30 9.62292
30 9.62 327
30 9. 62 362
30 9. 62 398
30
30
30
30
30
30
29
30
30
30
9.62 433
9. 62 468
9. 62 504
9. 62 539
9.62 574
9. 62 609
9.62645
9. 62 68Q
9. 62 715
9. 62 750
9. 62 785
10 cot
1570 2470 3370
'0
3510.38099 9.96535!
0.38 064 9.96 53Q
3610.38 028 9.96 525
36
35 0.37 992 9.96 520
3610.379579.96514
350.37921
9.96509
36 0.37886 9.96504
35'O.3785Q 9.96498
36: 0.37815 9.96493
3510.377799.96488
36 0.37744 9.96483
35' 0.37 708 9.96 477
35 0.37673 9.96472
3610.37 638 9.96 467
3510.37602 9.96461
3510.37567 9.96456
3610.37 532 9.96 451
35] 0.37 496 9.96 445
3510.37461
9.9644Q
3510.374269.96435
360.37391
9.96429
35 0.37 355 9.96 424
350.37320
9.96419
35 O.37 2~5 9.96 413
350.37250
9.96408
0.37215
9.95403
cd
log tan
log sin
67°
6
5
5
5
5
5
5
5
6
5
5
5
5
5
5
6
5
5
5
5
6
5
5
5
5
6
5
5
5
5
6
5
5
5
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
5
5
5
6
5
5
6
5
5
5
6
5
5
6
5
5
6
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
5
5
6
5
5
6
5
5
Id
TABLE III
i
8
7
6
5
4
3
2
I
0
I
1
2
3
4
5
6
7
8
9
10
20
30
40
50
37
0.6
1.2
1.8
2.5
3.1
3.7
4.3
4.9
5.6
6.2
12.3
18.5
24.7
30.8
36
0.6
1.2
1.8
2.4
3.0
3.6
4.2
4.8
5.4
6.0
12.0
18.0
24.0
30.0
36
0.6 .
1.2
1.8.':
2.3
2.9
3.5
4.1
4.7
5.2
5.8
11.7 ...
17.5
23.3
29.2
0
1
2
3
4
5
6
7
8
9
10
d
log tan
Ic d
18 9.59 720 29 9.63 414
9.59749
9.59 778
31
0.5
1.0
1.6
2.1
2.6
3.1
3.6
4.1
4.6
5.2
10.3
1
2
3
4
5
6
7
8
9
10
20
32
0.5
1.1
1.6
1.1
2.7
3.2
3.7
4.3
4.8
5.3
10 7
30
40
50
16.0 15.5
21.3 20.7
26.7 25.8
9.59 866
0.36
~~,
9.59924
9.59 954
29 9.63553
29 9.63 588
3510.36447
9.96284
0.36 412 9.96 278
~~I
0.36
1
2
3
4
5
6
7
8
9
29
0.5
1.0
1.4
1.9
2.4
2.9
3.4
3.9
4.4
6
O.I
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
5
0.1
0.2
0.2
0.3
0.4
0.5
0.6
0.7
0.8
10
20
30
40
50
4.8
9.7
14.5
19.3
24.2
1.0
2.0
3.0
4.0
5.0
0.8
1.7
2.5
3.3
4.2
9.60012
9.60 041
9.60 070
9.60 099
9.60 128
9.60157
9 60186
29
29
29
29
29
29
9.63761
9.63 796
9.63 830
9.63 865
9.63 899
9.63934
9.63968.
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
9.60 215 ! 29
9.60 244 29
9.60273 29
9.60 302 29
9.60 331 28
9.60 359 29
9.60 388
9.60417 29
29
9.60446 28
9.60 474 29
9.60 503 29
9.60532 29
9.60 561 28
9.60 589 29
9.60 618 28
9.60 646 29
9.60 675 29
9.64 003
9.64 037
9.64072
9.64 106
9.64 140
9.64 175
9.64 209
9.64243
9.64278
9.64 312
9.64 346
9.64381
9.64 415
9.64 449
9.64 483
9.64 517
9.64 552
52
53
54
55
56
57
58
59
60
Prop. Parts
88
166'
i
9.60 704
9.60 732
9.60 761
9.60789
9.60 818
9.60 842
9.60 875
9.60 903
9.60 93 I
28
29
28
29
28
29
28
28
log cos
d
2460 3360
9.64 586
9.64 620
9.64 654
9.64688
9.64 722
9.64 756
9.64 790
9.64 824
9.64 858
log cot
6
38
37
5
35
34
I
~36
9.96 256
~33
3510.36239
34 0.36 204
35 0.36 170
341 O.36 135
351 0.36 101
0.36066
34'
,10 36032
9.96251
9.96 245
9.96 240
9.96 234
9.96 229
9.96223'
21 I
3410.35 997
3510.35 963
34 0.35928
34 0.35 894
35 0.35 860
34 0.35 825
34 0.35 791
35 0.35757
34 0.35722
34 0.35 688
35 0.35 654
34 0.35619
34 0.35 585
34 0.35 551
34 0.35 517
35 0.35 483
34 0.35 448
9 96 212
9.96 207
9.96201
9.96 196
9.96 190
9.96 185
9.96 179
9.96174
9.96 168
9.96 162
9.96 157
9.96151
9.96 146
9.96 140
9.96 135
9.96 129
9.96 123
34
34
34
34
34
34
34
34
9.96 118
9.96 112
9.96 107
9.96101
9.96 095
9.96 090
9.96 084
9.96 079
9.96 073
6
5
6
6
5
6
5
6
log sin
d
Icd
0.35 414
0.35 380
0.35 346
0.35312
0.35 278
0.35 244
0.35 210
0.35 176
0.35 142
log tan
66°
I
2
3
4
5
6
7
8
9
10
20
30
40
50
36
0.6
i. 2
1.8
2.4
3.0
3.6
4.2
4.8
5.4
6.0
12.0
18.0
24.0
30.0
1
2
3
4
5
6
7
8
9
10
30
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
35
34
0.6
0.6
1.2
1.1
1.8
1.7
2.3
23
2.9
2.8
3.5
3.4
4. I
4.0
4.7
4.5
5.2
5. I
5.8
5.7
11.7 11.3
17.5 17.0
23.3 22.7
29.2 28.3
~39
9.96267
35
0.36 308 9.96 262
28
29
30
31
32
33
34
41
40
481 9.96 289
0.36274
~~I
49
~48
47
~46
45
~44
43
6
0.36 377 9.96 273
29 9.63 726
Prop. Parts
60
59
58
~57
56
55
~54
53
52
51
50
~42
9.96300
9.96 294
~~I
0.36343
30 9.63657
29 9.63 692
27 9.59983
586 9.96 305
3510.36551
O.36 516
24 9.59 895 29 9.63 623
25
26
10;6cos
29 9.63449
30 9.63 484
21 9.59 808 29 9.63 519
22 9.59837
23
log cot
d
3510.37215
9.964:>3
0.37 180 9.96 397 6
351. 0.37 145 9.96 392 5
~~10.37110 9.96387
9.96381
35 0.37074
5
0.37039
9.96376
0.37 004 9.96 370
~~I
35 0.36 969 9.96 365 5
0.36 934 9.96 360
6
9.96354
351 0.36899
34
5
35 ! 0.36 865 9.96 349 6
0.36 830 9.96 343
0.36 795 9.96 338
~~I
.0.36760
9.96333
0.36 725 9.96 327
~~I
i 0.36 690 9.96 322
~~I
0.36655
9.96 316
10.36621
9.96311
9.5:> 188
9.59 218
9.59 247
9.59277
9.59307
9.59336
9.59 366
9.59 396
9. 59 42~
9.59455
9.59 484
30 9.62785
820
29 9.62
9.62 855
~g9.62890
29 9.62926
9.62961
~g 9.62 996
29 9.63 031
30 9.63 066
29 9.63101
9.63 135
II 9.59 514 30 9.63 170
12 9.59 543
~~9.63205
13 9.59573
9.63240
2
14 9.59 602
3~ 9.63 275
15 9.59 632
9.63 310
16 9.59 661
~~9.63345
17 9.59690
309.63379
19
20
1130 2030 2930
23°
bgsin
Prop. Parts
d
6
5
6
5
I
32
31
30
~29
28
27
30
25
~24 40
23 50
I
29
0.5
1.0
1.4
1.9
2.4
2.9
3.4
3.9
4.4
4.8
28
0.5
0.9
1.4
1.9
2.3
2.8
3.3
3.7
4.2
4.7
I5. 0 14.5 I4 0
I
20. 0 19.3 18. 7
25.0 24.2 23.3
~22
5
5
6
5
6
21
20
~19
18
~17
16
15
14
~13
12
11
10
~9
8
7
6
5
4
3
2
1
0
1
2
3
4
5
6
7
8
9
10
20
30
40
50
6
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
5
O.I
0.2
0.2
0.3
0.4
0.5
0.6
0.7
0.9
1.0
2.0
3.0
4.0
5.0
0.8
0.8
1.7
2.5
3.3
4.2
'
89
24°
TABLE III
logsin fdll.Og tan I c d I logcot I logcos I d
'I
0 9.60931 29 9.64858 34 0.35142 9.96073 6
1 9.60 960 28 9.64 892 34 0.35 108 9.96 067 5
1 9.60988 28 9.64 926 34 0.35074 9.96062 6
3 9.61 01~ 29 9.64 960 34 0.35040 9.96056 6
4 9.61045 28 9.64994 34 0.35006 9.96 05~ 5
6 9.61073 28 9.65028 34 0.34972 9.96045 6
6 9.61 101 28 9.65 062 34 0.34 938 9.96 039 5
7 9.61 129 29 9.65096 34 0.34904 9.96034 6
8 9.61 158 28 9.65 130 34 0.34 870 9.96 028 6
9 9.61 186 28 9.65 164 33 0.34 836 9.96 022 5
10 9.61 214 28 9.65 197 34 0.34 803 9.96 017 6
11 9.61242 289.65231
340.34762
9.96011 6
12 9.61 270 28 9.65 265 34 0.34 735 9.96 005 5
13 9.61298 28 9.65299 34 0.34701 9.96000 6
14 9.61326 28 9.65333 33 0.34667 9.95994 6
16 9.61 354 28 9.65 366 34 0.34 634 9.95 988 6
16 9.61 382 29 9.65 400 34 O.34 600 9.95 982 5
17 9.61411 27 9.65434 33 0.34566 9.95977 6
18 9.61438 28 9.65467 34 0.34533 9.95971 6
19
9.61 466
20
21
22
23
24
125
26
27
28
29
30
31
32
9.61494
9.61 522
9.61 550
9.61578
9.61 606
9.61 634
9.61 662
9.61 689
9.61 717
9.61 745
9.61 773
9.61 800
9.61828
28 9.65 50
28
28
28
28
28
28
27
28
28
28
27
28
28
33 ~.6! ~~~I 27
,.
u'
'L~'
~
34 0.34 499
33 0.34465
34 O.34 432
34 0.34 398
33 0.34364
34 0.34 331
33 0.34 297
34 0.34 264
33 0.34 230
34 0.34197
33 0.34 163
34 0.34 130
0.34 096
33
34,0.34063
~.?~ 2~~I 331 ~.~~ 2~~
9.65535
9 65 568
9.65 602
9.65636
9.65 669
9.65 703
9.65 736
9.65 770
9.65803
9.65 837
9.65 870
9.65 9(\A\
9.65931
i ytj
9.95 965
'114° 204° 294°
Prop. Parts
60
59
58
57
56
66
54
53
52
51
60
49
48
47
46
45
44
43
42
40
39
38
37
36
35
34
33
32
31
30
29
28
27
.
36 9.61911 28 9.66038! 3310.33962 9.95 8731 ~ 25
36 9.61939 27 9.66071 33 0.33929 9.95868 6 24
1
37
38
39
40
41
42
43
44
45
46
47
48
49
60
51
52
53
54
66
56
57
58
59
60
9.61 966
9.61994
9.62 021
9.62049
9.62076
9.62104
9.62131
9.62159
9.62186
9.62214
9.62241
9.62268
9.62 296
9.62323
9.62350
9.6237Z
9.62405
9.62432
9.62459
9.62 486
9.62513
9.62541
9.62568
9.62595
28 9.66 104
27 9.66138
28 9.66 171
27 9.66204
28 9.66238
27 9.66271
28 9.66304
27 9.66337
28 9.66371
27 9.66404
27 9.66437
28 9.66470
27 9.66 503
27 9.66537
27 9.66570
28 9.66603
27 9.66636
27 9.66669
27 9.66702
27 9.66 735
28 9.66768
27 9.66801
9.66834
27
9.66867
34 0.33 896
33 0.33862
33 0.33 829
34 0.33796
33 0.33762
33 0.33729
33 0.33696
34 0.33663
33 0.33629
33 0.33596
33 0.33563
33 0.33530
34 0.33 497
33 0.33463
33 0.33430
33 0.33397
33 0.33364
33 0.33331
33 0.33 29~
33 0.33 265
33 0.33232
33 0.33199
33 0.33166
0.33133
9.95 862
9.95856
9.95 850
9.95844
9.95839
9.95833
9.95827
9.95821
9.95815
9.95810
9.95804
9.95798
9.95 792
9.95786
9.9578Q
9.95775
9.95769
9.95763
9.95757
9.95 751
9.95745
9.95739
9.95733
9.95728
6
6
6
5
6
6
6
6
5
6
6
6
6
6
5
6
6
6
6
6
6
6
5
102: sin
d
n
1660 2450 3360
65°
34
0.6
1.1
1.7
2.3
2.8
3.4
4.0
4.5
5.1
5.7
11.3
17.0
22. 7
28.3
1
2
3
4
5
6
7
8
9
10
20
30
40
50
33
0.6
1.1
1.6
2.2
2.8
3.3
3.8
4.4
5.0
5.5
11.0
16.5
22.0
27.5
41
5
9.95960 6
9.95 954 6
9.95 948 6
9.95942 5
9.95 937 6
9.95 931 6
9.95 925 5
9.95 920 6
9.95914 6
9.95 908 6
9.95 902 5
9.95 897 6
9.95891 6
2'2~ ~~~ I 6
.
log sin
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
27
0.4
0.9
1.4
1.8
2.2
2.7
3.2
3.6
4.0
4.5
<LJl
14.0 13.5
18.7 18.0
24.2 23.3 22.5
29
1
0.5
2
1.0
3
1.4
4
1.9
5
2.4
6
2.9
7
3.4
8
3.9
9
4.4
I
10
4.8
70 I 9 7
30 ! 14.5
40 19.3
50
1
1
2
3
4
5
6
7
8
9
10
20
30
40
50
6
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
2.0
3. 0
4.0
5. 0
28
0.5
0.9
1.4
1.9
2.3
2.8
3.3
3.7
4.2
4.7
6
0.1
0.2
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.8
1.7
2.:>
3.3
4. 2
0
\
2
3
4
5
6
7
8
9
10
11
12
13
14
16
t6
17
18
19
20
21
22
23
24
26
26
27
28
29
30
3\
32
33
34
35
36
37
38
39
40
41
42
43
44
46
46
47
48
49
60
51
52
53
54
65
56
57
58
59
60
Id I
log tan
90
Icd I
log cot
log cos
1
Icd
I
log tan
Prop. Parts
d
9.62595 27 9.66867 33 0.33133 9.95 728
9.62622 27 9.66900 33 0.33100 9.95 722 ~
9.62 649 27 9.66 933 33 0.33 067 9.95 716 6
9.62676 27 9.66966 33 0.33034 9.95710 6
9.62 703 27 9.66 999 33 0.33 00I 9.95704 6
9.62 730 27 9. 67 03~ 33 0.32 968 9.95 698 6
9.62 757 27 9.67 065 33 0.32 935 9.95 692 6
9.62 784 27 9.67 098 33 0.32 902 9.95686 6
9.62 811 27 9.67 131 32 0.32 869 9.95680 6
9.62 838 27 9.67 163 33 0.32 837 9.95 674 6
9.62 865 27 9.67 196 33 0.32 804 9.95 668 5
9.62 892 26 9.67 229 33 0.32 771 9.95663 6
9.62918 27 9.67 26~ 33 0.32738 9.95657 6
9.62 945 27 9.67 295 32 0.32 705 9.95 651 6
9.62 972 27 9.67 327 33 0.32 673 9.95645 6
9.62999 27 9.67360 33 0.32640 9.95 639 6
9.63 026 26 9.67 393 33 0.32 607 9.95 633 6
9.63052 27 9.67426 32 0.32574 9.95627 6
9.63 079 27 9.67 458 33 0.32 542 9.95621 6
9.63 106 27 9.67 491 33 0.32 509 9.95 615 6
9.63 133 26 9.67 524 32 0.32 476 9.95609 6
9.63159 27 9.67556 33 0.32444 9.95 603 6
9.63186 27 9.67589 33 0.32411 9.95 597 6
9.63 213 26 9.67 622 32 0.32 378 9.95591 6
9.63 239 27 9.67 654 33 0.32 346 9.95 585 6
9.63 266 26 9.67 687 32 0.32 313 9.95 579 6
9.63292 27 9.67719 33 0.32281 9.95 573 6
9.63 319 26 9. 67 75~ 33 0.32 248 9.95 567 6
9.63345 27 9.67785 32 0.32215 9.95561 6
9.63372 26 9.67 8~7 33 0.32183 9.95555 6
9. 63 39~ 27 9.67 850 32 0.32 150 9.95 549 6
9.63425 26 9.67 88~ 33 0.32118, 9.95543 6
9.63451 27 9.67915 32 0.32085 9.95 537 6
9.63478
9.67947
0.32053 9.95531
9.635041;~ 9.679801 ~~iO.32020 9.95 525 I ~
9.63 531 26 9.68 012 i 32' 0.31 988 9.9, 519 I 6
9.63 557 26 9.68 044 33' 0.31 956 9.95513'624
9.63 583 27 9.68 077 32 0.31 923 9.95 507 7
9.63610 26 9.68109 33 0.31891 9.95 500 6
9.63 636 26 9.68 142 32 0.31 858 9.95 494 6
9.63 662 27 9.68 174 32 0.31 826 9.95 488 6
9.63 682 26 9.68 206 33 0.31 794 9.95 482 6
9.63715 26 9.68239 32 0.31761 9.95 476 6
9.63741 26 9.68271 32 0.31729 9.95 470 6
9.63 767 27 9.68 303 33 0.31 697 9.95 464 6
9.63 794 26 9.68 336 32 0.31 664 9.95 458 6
9.63 820 26 9.68 368 32 0.31 632 9.95452 6
9.63846 26 9.68400 32,0.31600 9.95446 6
9.63 872 26 9. 68 43~ 33 0.31 568 9.95 440 6
9.63 898 26 9.68 465 32 0.31 535 9.95434 7
9.63 924 26 9.68 497 32 0.31 503 9.95 427 6
9.63 950 26 9.68 529 32 0.31 471 9.95421 6
9.63976 26 9.68561 32.0.31439 9.95415 6
0.31 407 9.95 409 6
9.64 002 26 9.68 593
9.64028 26 9.68626 33'
32 0.31374 9.95 403 6
9.64 054 26 9.68 658 32 0.31 342 9.95 397 6
9.64 080 26 9.68 690 32 0.31 310 9.95 391 7
9.64 106 26 9.68 722 32 0.31 278 9.95384
6
9.64 132 26 9.68 754 32 0.31 246 9.95 378 6
9.64 158 26 9.68 786 32 0.31 214 9.95 372 6
9. 64 184
9 . 68 818
O. 31 182 9.95 366
log cot
Prop. Parts
1160 2050 2960
25°
TABLE III
60
59
58
57
56
56
54
53
52
51
60
49
48
47
46
45
44
43
42
41
40
39
38
37
36
36
34
33
32
31
30
29
28
27
26
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
1
2
3
4
5
6
7
8
9
10
20
30
40
50
33
0.6
1. 1
1.6
2.2
2.8
3.3
3.8
4.4
5.0
5.5
11.0
16. 5
22.0
27.5
32
0.5
1.1
1.6
2. I
2.7
3.2
3.7
4.3
4.8
5.3
10.7
16. 0
21.3
26.7
27
0.4
0.9
1.4
1.8
2.2
2.7
3.2
3.6
4.0
4.5
26
0.4
0.9
1.3
1.7
2.2
2.6
3.0
3.5
3.9
4.3
9.0
8.7
1
2
3
4
5
6
7
8
9
10
20
30
40
50
1
2
3
4
5
6
7
8
9
10
20
30
40
50
I
i
1
7
0.1
0.2
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.2
2.3
3.5
4.7
5.8
13 5 13.0
18.0 17.3
22.5 21.7
6
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
t .0
2.0
3.0
4.0
5.0
Prop. parta
6
0.1
0.2
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.8
1.7
2.5
3.3
4.2
260
TABLE III
log
I I
sin
d
0
I
2
3
4
6
6
7
8
9
10
11
12
13
14
16
16
17
18
19
20
21
22
23
24
26
26
27
28
29
30
31
32
33
34
36
36
37
38
39
40
41
42
43
44
46
46
,47
48
49
60
51
52
53
54
66
56
57
58
59
9.64 184
9.64 210
9.64 236
9.64 262
9.64 288
9.64 313
9.64339
9.64 365
9. 64 391
9.64 417
9.64 442
9.64 468
9. 64 494
9.64 51 ~
9.64 545
9.64571
9.64 596
9.64 622
9.64647
9.64673
9.64698
9.64 724
9.64749
9.64 77;
9.64 800
9.64 826
9.64851
9.64 877
9.64 902
9.64927
9.64953
9.64978
9.65 003
9.65 029
9.65 054 iI
9.650791
9.65 104
9.6513Q
9.65 155
9.65 180
9.65205
9. 65 230
9.65255
9.65281
9.65 306
9.65331
9.65 356
9.65381
9.65 406
9.65431
9.65 456
9.65481'
9.65 506
9.65531
9.65556
9.65 580
9.65 605
9.65 630
9.65 655
9.65 680
60
9.65
70;
log cos
log
I cd I
tan
26 9.68
26 9.68
26 9.68
26 9. 68
25 9.68
26 9.68
269.69010
818
850
882
914
946
978
26 9.69 042
26 9.69 074
25 9. 69 106
26 9.69 138
26 9.69 170
25 9.69 202
26 9.69 234
26 9. 69 266
25 9.69298
9.69 329
26 9.69 361
25
9.69 39Z
26
259.69425
26 9.69457
9.69 488
25
26 9.69520
9.69 552
25
26 9.69 584
25 9.69 615
9.69647
26
25 9.69 679
25 9. 69 710
26 9.69742
25 9.69774
25 9.69805
26 9. &9 837
9.69 868,
25
25 9.69 900 i
9.69932 i
25
26 9.69 96z'
9.69995
25
25 9.70026
25 9.70 058
9.70089
25
25 9. 70 121
9.70152
26 9.70184
25
9.70 215
25
25 9.70247
25 9.70 278
25 9.70309
25 9. 70 341
9.70372
25
404
25 9.70
9.7043;
25
25 9.70 466
25 9.70498
24 9.70529
25 9.70 560
25 9.70 592
9.70 623
25 9.70
654
25 9. 70 685
25
, 9.70
I d I log
717
cot
log
1160 2060 2960
log cos
cot
j
d
32 O. 31 1~2 9.95366
6
32 0.31 150 9.95 360 6
32 0.31 118 9.95 354 6
32 0.31 086 9.95 348 7
32 0.31 054 9.95341
6
0.31 022 9.95 335
32
320.30990
9.95 329 6
6
32 0.30 958 9.95 323 6
O.
30
926
9.95317
32
7
32 0.30 894 9.95310
6
0.30
862
9. 95 304 6
32
32 0.30 830 9.95298
O.
30
798
9.95
292 6
32
6
32 0.30 766 9.95286
7
32 0.30 734 9.95 279 6
9.95 273 6
31 0.30702
32 0.30 671 9.95 267 6
32 0.30 639 9.95261
7
0.30607
9.95 254 6
32
320.30575
9.95 248 6
9.95 242 6
31 0.30543
32 0.30 512 9.95 236 7
0.30480
9.95229
32
6
32 0.30 448 9.95 223 6
9.95 217
31 O. 30 41
6
32 0.30 385 9.95211
7
0.30353
9.
95
204
32
6
0.30
321
9.95198
31
32 O. 30 290 9.95 192 ~
9.95185
32 0.30258
6
9.95179
31 0.30222
6
9.95 173 6
32 0.30195
9.95
167
31 0.30 163
I 7
32! 0.30 132 9.95 160, 6
321 0.30 100 9.95 154 I 6
9. 95 148 i 7
31' 0.30068
32 0.30 037 9.95141'
6
9.95 135 6
31 0.30005
9.95129
0.29
974
32
7
31 0.29 942 9.95 122 6
9.95
116
0.29911
32
6
31 0.29 879 9.95 110 7
9.95103
32 0.29848
6
9.95097
31 0.29812
7
32 0.29 785 9.95090
6
9.95
084
0.29753
31
6
31 0.29 722 9.95 078 7
32,0.29691
9.95071
6
31 0.29 659 9.95065
6
9.95
059
0.29628
32
7
31 0.29 596 9.95 052 6
9.95046
31 0.29565
7
32 0.29 534 9.95039
6
0.29502
9.95
033
31
9.95 027 6
31 0.29471
7
0.29
440
9.95 020
32
0.29 408 9.95 014 6
31 0.29 377
7
31 0.29 342 9.95007
6
9.95001
31 0.29 31 5 9. 94 995 6
32
7
0.29 283 9.94 988
I cd
~
log
tan
log sin
Id
Prop. Parts
60
59
58
57
56
65
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
I
0
32
1
0.5
2
1.1
31.61.6
4
2. I
5
2.7
6
3.2
7
3.7
8
4.3
9
4.8
10
5.3'
20 10.7
30 16. 0
40 21.3
50 26.7
270
TABLE III
log sin
31
0.5
1.0
2. I
2.6
3.1
3.6
4. I
4.6
5.2
10.3
15 . 5
20.7
25.8
d
log tan
cd I
log cot
I
1170 2070 297°
log cos
Id I
0 9.65 705 24 9. 70 717 31 0.29 283 9.94 988 I 6
I 9.65 729 25 9 70 748 31 0.29 252 9 94 982 7
2 9 65 754 25 9.70 779 31 0.29 221 9.94 975 6
9. 9~' 969 7
3 9.65 779 25 9.70 810 31 0 29190
4 9. 65 804 24 9. 70 841 32 0.29 159 9. \#,. 962 6
9.94956
6 9.65 828 25 9.70 873 31 0.29127
7
6 9.65 853 25 9.70 901 31 0.29 096 9.94 949 6
7 9. 65 8~8 24 9.70 935 31 O. 29 065 9. 94 943 7
8 9.65 902 25 9.70 966 31 0.29 034 9.94 936 6
9 9.65 927 25 9.70 997 31 0.29 003 9.94 930 7
9.94923
10 9.65 952 24 9.71 028 310.28972
6
O. 28 941 9. 94 91 7
11 9.65976
25 9.71 059 3 I 0 28 910 9. 94 911 6
12 9.66001
7
31
24 9.71090
320.28879
9.94904
13 9.66025
259.71121
6
14 9.66 050 25 9.71 153 31 0.28 847 9.94 898 7
15 9.66 075 24 9.71 181 31 0.28 816 9.94 891 6
16 9.66 099 25 9. 71 215 31 0.28 785 9.94 885 7
0.28754
9,94878
17 9.66124
7
24 9.71246
31
18 9 66 148 25 11.71 277 310287239.94871642
19 9. 66 173 24 9. 71 308 31 0.28 692 9.94 865 7
20 9.66197
31 0.28 661 9.94 858 6
24 9,71339
21 9.66221
9.94852
7
25 9.71 370 3110.28630
22 9.66246
30 0.28 599 9 94 845 '6
24 9.71401
23 9.6627Q
31 0.28 569 9 94 839 7
25 9.71431
249.66295249.71462
31 0.28 538 9.94 832 6
25 9.66 319 24 9.71 493 31 O. 28 5,}7 9.94 826 7
26 9.66343
6
3110 28 47~ 9.94819
25 9.71524
3110.284459.9481317
27 9.66368
24 9.71555
28 9.66 392 24 9.71 586 31 O. 28 414 9 . 94 806' 7
9.94799
29 9.66416
6
31 0.28383
25 9.71617
9.71648
310.28352
994793
30 9.66441
7
24
300.28321
9.94786
31 9.66465
6
24 9.71679
9.94
780
32 9.66 489 24 9.71 709 31' 0.28 29 i
7
33 9.66513
31! 0.28 260 9.94 773 ! 6
24 9,71740
34 9.66537
25 9.71771
31' 0 28 229 9 94 767 I 7
9.947601
35 9. 66 562 24 9.71 802 3110.23198
7
36 9.66 586 24 9.71 833 301 0.28 167 9.94 753 6
0.28137
9.94747
9.71
863
.37 9.66610
7
31
24
138 9.66 634 24 9. 71 891 31 0, 28 106 9. 94 740 6
39 9.66 658 24 9. 71 925 30 0.28 075 9.94 734 7
310.28045
9.94727
40 9.66682
7
24 9.71955
41 9.66 706 25 9.7\ 986 31 0.28 014 9.94 720 6
42 9.66731
24 9.72 017 31 O. 27 983 9 . 94 714 7
300.27952
9.94707
43 9.66755
249.72048
7
44 9.66 779 24 9.72 078 31 O. 27 922 9. 94 700 6
9.94694
45 9.66 803 24 9.72 109 31 0.27891
7
46 9.66 827 24 9.72 140 30 0.27 860 9.94 687 7
0.27
830
9.94
680
9.72
170
47 9.6685!
6
31
24
9.94674
48 9.66 875 24 9. 72 201 30 0.27799
7
0.27769
9.94667
9.72
231
49 9.66899
7
31
23
50 9.66 922 24 9.72 262 31, 0.27 738 9.94 660 6
51 9.66 946 24 9. 72 293 300.27707
9,94654
7
52 9.66 970 24 9. 72 323 31 0.27 677 9.94 647 7
53 9.66994
30, 0.27 646 9.94 640 6
24 9.72354
9.94634
54 9.67 018 24 9.72 384 31 0.27616
7
55 9.67 042 24 9.72 415 30 0.27 582 9.94 627 7
56 9.67 066 24 9.72 445 31' 0.27 555 9.94620
6
57 9.67090
30! 0.27 524 9.94614
7
23 9.72476
58 9.67 113 24 9. 72 506 311 0.27 494 9.94 607 7
599.67137249.72537
30, 0.27 463 9.94 600 7
60 9.67 161
9.72 567
0.27 433 9.94 593
1
1
2
3
4
5
6
7
8
9
10
20
30
40
50
'
I
26
0.4
0.9
1.3
1.7
2.2
2.6
3.0
3.;
3.9
4.3
8. 7
13. 0
I7.3
21 .7
I
2
3
4
5
6
7
8
9
IJ
20
30
40
50
7
0.1
0.2
0.4
0.;
0.6
0.7
0.8
0.9
1.0
1.2
2.3
3. 5
4. 7
5 .8
25
0.4
0.8
1.2
1.7
2.1
2.5
2.9
3.3
3.8
4.2
8. 3
12 . 5
16 . 7
20.8
6
O. I
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
I. 0
2. 0
3. 0
4. 0
5 .0
24
0.4
0.8
1.2
1.6
2.0
2.4
2.8
3.2
3.6
4.0
8.0
12.0
16.0
20.0
1
1
.
I
1
1
1
Prop. Parts
I
log tan
I
62°
log sin
Prop. Parts
60
59
58
57
56
55
54
53
52
51
60
49
48
47
46
45
44
43
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
I6
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
Id I ' I
32
31
30
0.5
0,5
0.5
2
1.1
3161.61.5
42.12.12.0
52.7262.5
63.23.130
7
3.7
84.34.14.0
9
4.8
10
5.3
20 10. 7
30 16.0
40 21. 3
50 26.7
1.0
1.0
3.6
3.5
1
1
25
0.4
0.8
1.2
1.7
2.1
2.5
1
2
3
4
5
6
7
2.9
8
3.3
9
3.8
10
4.2
20
8 3
30 i 12.5
40 ' 16.7
50 20.8
I
46
4.5
5.2
5.0
I 0 . 3 . 10. 0
15.5 15.0
20. 7 20. 0
25.8 25.0
24
0.4
0.8
1.2
I. 6
2.0
2.4
2.8
3.2
3.6
4.0
8 0
12.0
16.0
20.0
7
1 0.1
2 0.2
3 0.4
4 0.;
5 0.6
6 0.7
7 0.8
8 0.9
9 1.0
10 1.2
20 2 . 3
30 3 . 5
404.74.0
50 5.8
23
0.4
0.8
1.2
1.5
1.9
2.3
2.7
3.1
3.4
3.8
7.7
11.5
15.3
19.2
6
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
2 .0
3.0
5.0
Pro:>. Parts
93
l
TABLE III
28°
~Id I
0
I
2
3
4
6
6
7
8
9
10
II
12
13
14
16
16
17
18
19
9.67161
9.67 185
9.67 208
9.67 232
9.67 256
9.67 280
9.67303
9.67 327
9.67 350
9.67374
9.67 398
9.67421
9.67 445
9.67468
9.67 492
9.67 515
9.67 539
9.67562
9.67 586
9.67 609
9.67
633
120
21 9.67 656
22 9.67680
23 9.67703
24 9.67726
26 9.67750
26 9.67 773
27 9.67 796
28 9.67 820
29 9.67 843
30 9. 67 866
31 9.67 890
32 9.67913
33 9. 67 936
log tan
17dl
249.72567
23 9. 72 598
24 9.72 628
24 9.72 659
24 9.72 689
23 9. 72 720
24 9.72 750
23 9.72 780
24 9.72 811
24 9.72 841
23 9.72 872
24 9.72 902
23 9.72 932
249.72963
23 9. 72 993
24 9.73 023
23 9.73 054
24 9.73084
23 9.73 114
24 9. 73 144
23 9.73 175
24 9.73 205
23 9.73235
23 9.73265
24 9.73295
23 9.73326
23 9.73 356
24 9. 73 386
23 9.73 416
23 9.73 446
24 9.73476
23 9. 73 507
239.73537
23 9.73567
23
.
log cot
I
Id I
log cas
310.274339.94593
30 0.27 402 9.94 587
31 0.27 372 9.94 580
30 0.27 341 9.94 573
3 I 0.27 311 9.94 567
30 O.27 2~0 9.94 560
9.94553
30 0.27250
31 0.27 220 9.94 546
30 0.27 189 9.94 540
9.94533
31 0.27159
30 0.27 128 9.94 526
9.94519
30 0.27098
0.27 068 9.94 513
31
300.27037
9.94506
30 O. 27 007 9.94 499
31 0.26 977 9.94 492
30 0.26 946 9.94 485
9.94479
30 0.26916
30 0.26 886 9.94 472
31 0.26 856 9.94 465
30 0.26 825 9.94 458
30 O.26 79~ 9.94 451
30 0.26 76~ 9.94445
30 0.26 73~ 9.94438
9.94431
31 0.26705
9.94424
30 0.26674
30 0.26 644 9.94 417
0.26 614 9.94 410
301 0.26 584 9.94 404
30,
30 0.26 554 9.94 397
9.94390
31 0.26524
30 0.26 493 9.94 383
300.26463
9.94376
10.26433
I
9.94369
6
7
7
6
7
7
7
6
7
7
7
6
7
7
7
7
6
7
7
7
7
6
7
7
7
7
7
6
7
7
7
7
7
9.68006
9.68 029
9. 68 052
9.68075
139
40 9.68 098
,4\ 9.68121
42 9.68 144
43 9.68 167
44 9.68 190
46 9.68213
46 9.68237
47 9. 68 260
48 9.68 283
49 9.68 305
60 9.68 328
51 9.68351
52 9.68374
53 9.68 397
54 9.68420
66 9. 68 443
56 9. 68 466
57 9.68489
58 9. 68 512
59 9.68534
60 9.68557
I
239.73657
3010.263439.94349
7
23 9.73 687 30 0.26 313 9. 94 34~ 7
O.26 283 9 94 33S
23 9 73 717
7
9.94328
23 9.73747
301 0.26253
30
7
9.73
777
0.26
223
9.94
321
23
30
7
9.73807
3010.26193
9.94314
23
7
23 9.73 837 30 0.26 163 9.94 307 7
9.73
867
0.26
133
9.94
300
23
30
7
23 9. 73 897 30 0.26 103 9.94 293 7
24 9.73927
30 0.26 073 9. 94 286 7
9. 94 279 6
23 9.73957
30 0.26043
9.94273
23 9. 73 987 30 0.26013
7
22 9.74017
30 0.25 983 9.94 266 7
9.94 259 7
23 9.74047
30 0.25953
9.94 252 7
23 9.74 077 30 0.25923
9.94245
23 9.74107
30 0.25893
7
9.94 238 7
23 9.74137
29 0.25863
9.74166
0.25
834
9.
94
231
23
30
7
23 9.74196
30 O.25 804 9.94 224 7
9. 94 217 7
23 9.74226
30 0.25774
23 9.74256
30 0.25 744 9. 94 210 7
23 9. 74 286 30 0.25 714 9.94 203 7
22 9.74316
29 0.25 684 9. 94 196 7
9.94 189 7
23 9.74345
30 0.25655
9.74375
O.25 625 9. 94 182
logcos
1610 2410 3310
~log cot
I
60
59
58
57
56
66
54
53
52
51
60
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
30
9.67 982 I 24 9.73 627' 30: 0.26 373 9.94 355 '6
36
36
37
38
~log tan
I
logsin
610
Id I
TABLEm
118" 2080 2980
26
24
23
22
21
20
19
18
17
16
16
14
13
12
11
10
9
8
7
6
6
4
3
2
1
Prop. Parts
1
2
3
4
5
6
7
8
9
10
20
30
40
50
31
0.5
1.0
1.6
2.1
2.6
3.1
3.6
4.1
4.6
5.2
10.3
15.5
20.7
25.8
30
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
10.0
15.0
20.0
25.0
29
0 5
1.0
I 4
1 9
2 4
2.9
3.4
3.9
4.4
4 8
9.7
14.5
19.3
24.2
24
0.4
0.8
1.2
1.6
2.0
2.4
2.8
3.2
3.6
4.0
23
0.4
0.8
1.2
1.5
1.9
2.3
2.7
3.1
3.4
3.8
7 7
11.5
15.3
19.2
22
0.4
0.7
1.1
1.5
1.8
2.2
2.6
2.9
3.3
3.7.~
7.3
11.0
14.7
18.3
) 12.0
40 116.0
50 20. 0
1
2
3
4
5
6
7
8
9
10
20
30
40
50
7
0.1
0.2
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.2
2.3
3.5
4.7
5.8
0
1
2
3
4
6
6
7
8
9
10
II
12
13
14
15
16
17
18
19
20
21
22
23
24
26
26
27
28
29
30
31
32
33
6
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
2.0
3.0
4.0
5.0
01
'I
,
Prop. Parts
94
.
........
''''
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
60
51
52
53
54
66
56
57
58
59
60
29°
log sin
9 68 557
9.68580
9 68 603
9 68 625
9 68 648
9.68 671
9.68694
9.68716
9.68 739
9.68762
9 68 784
9.68 807
9.68829
9. 68 85~
9.68875
9 68 897
9 68 920
9.68 94~
9.68965
9.68987
9. 69 010
9.6903~
9.69055
9.69 077
9. 69 100
9. 69 122
9.69 144
9.69 167
9.69 189
9. 69 212
9.69 234
9.69 256
9. 69 279
~.6~ ~2!
,.
J~J
d
cd
1011"tan
1190 209' 2990
log cas
log cot
0 25 625 9 94 18~ I
n 9 74 375 '
~~0.25
595 9 94 175 I 77
23 9 74 40~
9. 94 168
9.74435130
0.25565
22
0 25 535 9. 94 161 7
23 9 74465
7
23 9.74 494
~6 0 25 506 9 94 154 7
9.74524
025476
9
94
147
23
~~0.25446 9. 94 140 7
22 9 74554
7
0 25 417 9.94133
23 9 74 583
7
23 9. 74 613 j~ 0.25 387 9. 94 126 7
300.25357
9. 94 I i 9
22 9.74643
7
9. 94 11
23 9.74673
~7
29 0.25327
9.74
702
0.25
298
9.94105
22
30
7
23 9.74 732 30 0.25 268 9 94 098 8
9.74762
0.25238
9.
94
090
23
29
7
22 9. 74 791 30 O.25 209 9. 94 083 7
9.74821
0.25179
9.94076
23
30
7
22 9. 74 851 29 0.25 149 9. 94 069 7
9.
74
880
0.25
120
9.
94
062
23
30
7
290.25090
9.94055
22 9.74910
7
9.74
939
0.25
061
9.
94
048
23
30
7
9. 94 041 7
22 9.74969
29 0.25031
23 9.74 998 30 0.25 002 9. 94 034 7
22 9.75 028 30 0.24 972 9.94027
7
290.24942
9.94 020 8
23 9.75058
9.75087
0.24913
9.
94
012
22
30
7
22 9.75 117 29 0.24 883 9.94 005 7
9.75
146
0.24854
9.93
998
23
30
7
9.93 991 7
22 9.75176
29 0.24821
9.75205
0.24795
9.93
984
23
30
7
290.24765
9.93 977 7
22 9.75235
22 9.75 264 30 0.24 736 9.93 970 7
23 9.75 294 29 0.24 706 9.93 9631 8
300.24677
9.93 955 7
22 9.75323
22 9.75 353 29, 0.24 647 9.93 948, 7
I
22
9.69345123
9.69 368 22
9.69 390 22
9.69 412 22
9. 69 434 22
9.69 456 23
9.69479
22
9.69 50 I 22
9.69 523 22
9.69545
22
9.69 567 22
9.69 589 22
9. 69 611 22
9.69 633 22
9.69 655 22
9.69 677 22
9. 61) 699 22
9.69721
22
9.69743
22
9. 69 765 22
9.69 787 22
9. 69 809 22
9. 69 831 22
9.69853
22
9.69875
22
9.69 897
log cas
d
I
0 2400 3300
7. I J JV~
29
v. £'f U I 0
9. 75 411 :
30' O.24 589
9.75441
29'0.24559
9.75170
3010.24530
9.75500
290.24500
9.75529
290.24471
9.75558
30 0.24442
9.75588
29 0.24412
9.75617
30 0.24383
9.75647
29 0.24353
9.75676
29 0.24324
9. 75 70~ 30 0.24 295
9.75 735 29 0.24 265
9.75764290.24236
9.75793
29 0.24207
9.75 822 30 0.24 178
9.75852
29 0.24148
9.75881
29 0.24 1\9
9.75910
29 .0.24090
9.75939
30 0.24061
9.75969
29 0.24031
9. 75 998 29 0.24002
9.76027
29 0.23 973
9.76056
30 O.23 944
9.76086
29 0.23914
9.76115
29 0.23885
9.76144
I 0.23 856
log cot
led
log tan
60
59
58
57
56
66
54
53
52
51
60
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
7
7.7J7"rI
9. 93 934
9.93 927 '7 7
9.93 920 8
9.93 912 7
9.93 905 7
9.93 898 7
9.93 89\ 7
9.93 884 8
9.93 876 7
9.93 869 7
9. 93 86~ 7
9.93 855 8
9.93847
7
9.93 840 7
9.93 833 7
9.93 826 7
9.93 819 8
9.93811
7
9.93 804 7
9.93 797 8
9.93789
7
9.93 782 7
9.93775
7
9.93768
8
9.93760
7
9.93753
log sin
d
I
0
Prop Parts
d
~v
25
24
23
22
21
20
19
18
17
16
16
14
13
12
II
10
9
8
7
6
6
4
3
2
1
0
'
1
2
3
4
5
6
7
8
9
10
20
30
40
50
30
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
10.0
15.0
20.0
25.0
29
0.5
1.0
1.4
1.9
2.4
2.9
3.4
3.9
4.4
4.8
9.7
14.5
19.3
24.2
I
2
3
4
5
6
7
8
9
10
23
0.4
0.8
1.2
1.5
1.9
2.3
2.7
3. I
3.4
3.8
22
0.4
0.7
1.1
1.5
1.8
2.2
2.6
2.9
3.3
3.7
£J
I. I
33 111.5
40 15.3
50 19.2
I
2
3
4
5
6
7
8
9
10
20
30
40
50
8
0.1
0.3
0.4
0.5
0.7
0.8
0.9
1.1
1.2
1.3
2.7
4.0
5.3
6.7
11.0
14.7
18.3
7
0.1
0.2
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.2
2.3
3.5
4.7
5.8
I
Prop. Parts
95
l
TABLE III
log sin
,
I
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
300
I d I logtan
9.69 897
9.69919
9. 69 941
9.69 963
9. 69 984
9. 70 006
9.70 028
9.70050
9.70 072
9.70093
9.70 115
9.70137
9. 70 159
9. 70 180
9.70 202
9. 70 224
9. 70 245
9.70 267
9. 70 288
9. 70 310
9.70 332
9. 70 353
9.70375
9. 70 396
9.70 418
9.70 439
9. 70 461
9.70 482
9. 70 504
9.70 525
9.70547
9. 70 568
9. 70 590
9.70611122
9.70633121
9. 70 654 i
9.70 675
9. 70 697
9.70718
9.70739
9.70761
9.70782
9.70 803
9.70 824
9.70 846
9. 7a 867
9. 70 888
9.70 909
9.70 931
9.70 952
9.70973
9.70 994
9.71 015
9.71036
9.71 058
9.71079
9.71 100
9.71 121
9.71 142
9.71 163
9.71 184
og cos
22
22
22
21
22
22
22
22
21
22
22
22
21
22
22
21
22
21
22
22
21
22
21
22
21
22
21
22
21
22
21
22
21
9.76 144
9.76173
9.76202
9.76231
9.76261
9.76290
9.76319
9.76348
9.76377
9.76406
9.76435
9.76464
9.76493
9.76522
9.76551
9. 76 580
9.76609
9.76639
9. 76 668
9.76697
9.76725
9.76 754
9.76783
9.76812
9.76841
9.76870
9.76899
9.76 928
9.76957
9.76 98~
9.77 015
9. 77 044
9.77073
9.77 101
9.77 130
0.23 856
0.23 827
291
29
29 0.23 798
3010.23769
29 0.23 739
29 0.23710
29
O. 23 681
29 0.23652
0.23623
O. 23 59~
291
29
29 0.23 565
29 0.23536
29 0.23507
29 0.23478
29 0.23449
290.23420
3010.23391
29 0.23 361
290.23332
28
O. 23 30~
29 0.23 275
29 0.23246
29 0.23 217
29 0.23188
29 0.23 159
29 0.23130
29 0.23 101
29 0.23 072
290.23043
29 0.23014
29 0.22 985
29 0.22 956
28 0.22927
! 29! 0.22 899
1 291 0.22 870
i
21 9.77 159 2910.22 841
22
21
21
22
21
21
21
22
21
21
21
22
21
21
21
21
21
22
21
21
21
21
21
21
d
1490 2390 3290
" 2100 300.
1200
led I logcot I logcos I d I
9.77 188 29 0.22 812
9.77217
290.22783
9.77 246 28 0.22 754
9.77 274 29 0.22 726
9. 77 303 29 0.22697
9. 77 332 29 0.22668
9.77361
29 0.22639
9. 77 390 28 O.22 610
9.77 418 29 0.22 582
9.77 447 29 0.22553
9.77476
29 0.22 524
9.77 505 28 0.22495
9.77533
29 0.22467
9.77562
29 0.22438
9.77591
28 0.22409
9.77 619 29 0.22381
9.77648
29 0.22352
9.77 677 29 0.22323
9. 77 706 28 0.22 294
9.77 734 29 O.22 266
9.77763
28 0.22237
9.77791
29 0.22209
9. 77 820 29 0.22180
9.77 849 28 0.22 151
O.22 123
9.77 877
log cot cd log tan
9.93753
7
9.93746
8
9.93738
7
9.93 731 7
9.93724
7
9.93717
8
9.93 709 7
9.93 702 7
9.93695
8
9.93687
7
9.93 680 7
9.93673
8
9.93 665 7
9.93 658 8
9.93 650 7
9.93643
7
9.93 636 8
9.93 628 7
9.93621
7
9.93614
8
9.93 606 7
9.93 599 8
9.93 591 7
9.93 584 7
9.93577
8
9.93 569 7
9.93 562 8
9.93 554 7
9.93547
8
9.93539
7
9. 93 53~ 7
9.93 525 8
9.93 517 7
9.9351018
9.93 502 I 7
9.93495 i 8
9.93 487 7
9 . 93 480 8
9.934n
7
9.93 465 8
9.93457
7
9.93450
8
9.93 442 7
9.93435
8
9.93427
7
9.93420
8
9.93 412 7
9.93405
8
9.93 397 7
9.93 390 8
9.93 382 7
9.93375
8
9.93 367 7
9.93 360 8
9.93 352 8
9.93 344 7
9.93 337 8
9.93 329 7
9.93322
8
9.93 314 7
9.93 307
log sin I d
o9Q
I
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
'
1
2
3
4
5
6
7
8
9
10
20
30
40
50
30
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
10.0
15.0
20.0
25.0
1
2
3
4
5
6
7
8
9
10!
20 1
30
29
0.5
1.0
1.4
1.9
2 4
2.9
3 4
3 9
4 4
4.8
9.7
14.5
19.3
24.2
22
0.4
0.7
1.1
1.5
1.8
2.2
2.6
2.9
3.3
3.7
7.3
i 11. 0
40 114.7
50 18.3
1
2
3
4
5
6
7
8
9
10
20
30
40
50
.
I
8
0.1
0.3
0.4
0.5
0.7
0.8
0.9
1.1
1.2
1.3
2.7
4.0
5.3
6.7
28
0.5
0.9
1.4
1.9
2.3
2.8
3.3
3.7
4.2
4.7
9 3
14.0
18.7
23.3
21
0.4
0.7
1.0
1.4
1.8
2.1
2.4
2.8
3.2
3.5
7.0
10.5
14.0
17.5
-...
7
O. J
0.2
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.2
2.3
3.5
4.7
5.8
Prop. Parts
90
3r
TABLE III
,
Prop. Parts
.
0
t
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
log sin
d
9.71 181
9.71205
9.71226
9.71247
9. 71 268
9.71289
9.71310
9.71331
9.71 352
9.71373
9.71393
9.71414
9.71435
9.71456
9.71477
9. 71 498
9.71 519
9.71539
9.71560
9.71581
9.71 602
9. 71 622
9.71643
9. 71 661
9.71685
9.71 705
9.71 726
9.71747
9.71767
9.71 788
9. 71 809
9. 71 8~9
9.71850
9.71870!21
9 71 891 I
9.71 911 !
9. 71 932
9.71952
9.71973
9. 71 994
9. 72 014
9.72 031
9.72 055
9.72 075
9. 72 096
9.72 116
9.72 137
9.72 157
9.72 177
9 . 72 198
9.72218
9.72 238
9.72 259
9.72 279
9.72 299
9.72 320
9. 72 340
9. 72 360
9. 72 381
9.72401
9.72421
log cos
1'li1" ..j-o'
I
21
21
21
21
21
21
21
21
21
20
21
21
21
21
21
21
20
21
21
21
20
21
21
21
20
21
21
20
21
21
20
21
20
20
21
20
21
21
20
20
21
20
21
20
21
20
20
21
20
20
21
20
20
21
20
20
21
20
20
Id I
~~6
log tan
log cot
cd
log cos
9. 77 877
0.22 123
9. 77 90§ 29 0.22 094
29 0.22065
9.77935
28
9.77 963 29 0.22037
9.77 992 28 0.22 008
9. 78 020 29 0.21 980
9. 78 049 28 0.21 951
9.78 077 29 0.21 923
9.7810§
29 0.21894
9.78135
28 0.21865
9.78163
29 0.21837
9. 78 192 28 O.21 8a8
9.78 220 29 0.21 73a
9.78 249 28 0.21 751
9.78 277 29 0.21 723
9.78 306 28 0.21 694
9.78 334 29! 0.21 666
9.78363
28!0.21637
9.783912810.21609
9.78419
29 0.21 581
9.78448
2810.21552
9.7847§
29'0.21524
9.785:)5
2810.21495
9.78 533
0.21 467
291
9.78562
28 0.21438
9.78590
2810.21410
9.78618
29 0.21382
9.78647
28 0.21 35~
9.78 675
0.21 325
9.78 704 291
28 0.21 296
9. 78 732 28; 0.21 268
9.78 760
0.21 240
291
9.78789
28,0.21211
!}.7~817! 28! 0.21 183
9.78 845 1 291 0 21 155
9.788741 2810.21 126
9.78902
28 0.21098
9.78930
29 0.21070
9.78959
28 0.21041
9.78 987 28 0.21 013
9.79 015 28 0.20 985
9.79043
29 0.20957
9.79 072 28 0.20 928
9. 79 100 28 O.20 900
9.79128
28 0.20872
9.7') 156 29 0.20844
9. 79 185 28! 0.20815
9. 79 213 281 0.20787
9. 79 241
0.20759
28'
9.79269
2810.20731
9. 7'J 297 29 0.20703
9.79 326
0.20674
281 0.20646
9.79 354 28
9. 79 382 28 O.20 618
9. 79 410 28 0.20590
9.79438
28 0.20562
9. 79 466 29 0.20534
9.79495
28 0.20505
9.79523
28 0.20477
9.79551
28 0.20449
9.79579
0.20421
tog cot
cd
I
log tan
1210 21103010
d
9.93 3a7
9.93 299 8
9.93 291 8
9.93 234 7
9.93 276 8
7
9.93 269
8
9.93261
8
9.93253
9.93 246 7
8
9.93238
8
9.93230
9.93 223 7
9.93 215 8
9.93 207 8
7
9.93 200 8
9.93192
8
9.93 184 7
9.93 177 8
9.93 169 8
9.93 161 7
9.93 154 8
9.93146
8
9.93 138 7
9.93 131 8
9.93 123 8
9.93 115 7
9.93108
8
9.93 100 8
9. 93 092 8
9.93 084 7
9.93 077 8
9.93 069 8
9.93061
8
9.93 053 ! 7
9 93 046 I 8
I
9 93038 8
I
9.93030
9.93 022
9.93014
9.93 007
9. 92 999
9.92 991
9.92 983
9.92 976
9. 92 968
9.92 960
9.92952
9. 92 944
9.92 936
9.92 929
9.92921
9.92913
9.92 905
9.92 897
9.92 889
9.92 881
9.92874
9. 92 866
9.92 858
9.92 850
9.92842
8
8
7
8
8
8
7
8
8
8
8
8
7
8
8
8
8
8
8
7
8
8
8
8
log sin!
d
---w
Prop. Parts
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
I
29
0.5
1.0
1.4
1.9
2.4
2.9
3.4
3.9
4.4
4.8
9.7
14.5
19.3
24.2
28
0.5
0.9
1.4
1.9
2.3
2.8
3.3
3.7
4.2
4.7
9.3
14.0
18.7
23.3
21
1
0.4
2
0.7
3
1.0
4
1.4
5
1.8
6
2.1
7
2.4
8
2.8
9
3.2
IO! 3.5
20 1 7 0
30 jl0. 5
40 14.0
50 17.5
20
0.3
0.7
1.0
1.3
1.7
2.0
2.3
2.7
3.0
3.3
6 7
10.0
13.3
16.7
1
2
3
4
5
6
7
8
9
10
20
30
40
50
1
2
3
4
5
6
7
8
9
10
20
30
40
50
8
0.1
0.3
0.4
0.5
0.7
0.8
0.9
1.1
1.2
1.3
2.7
4.0
5.3
6.7
Prop.
7
0.1
0.2
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.2
2.3
3.5
4.7
5.8
Parts
m
log sin
I
320
TABLE III
d
log tan
cd
log cot
1220 212" 3020
log cas
d
I
0 9.72421 209.79579
280.20421
9.92842 8 60
1 9.72441 20 9.79607 28 0.20392 9.92834 8 59
2 9.72 461 21 9.79 635 28 0.20 365 9.92 826 8 58
3 9.72482 209.79663
280.20337
9.92818 8 57
4 9.72 502 20 9.79 691 28 0.20 309 9.92 810 7 56
5 9.72 522 20 9.79719 28 0.20281 9.92802 8 55
6 9.72542 209.79747
290.20253
9.92795 8 54
7 9.72562 20 9.79776 28 0.20224 9.92787 8 53
8 9. 72 582 20 9. 79 804 28 0.20 196 9.92 779 8 52
9 9.72 602 20 9.79832 28 0.20168 9.92 771 8 51
10 9.72 622
9.79 860
0.20 140 9.92 762 8 50
~~0.20
112 9.92 755 8 49
11 9.72 643 ~69.79 888
12 9.72 663 20 9.79916 28 0.20084 9.92 747 8 48
13 9.72 683 20 9.79 944 28 0.20 056 9.92 739 8 47
14 9.72 703 20 9.79 972 28 0.20 028 9.92 731 8 46
15 9.72 723 20 9.80 000 28 0.20 000 9.92 723 8 45
16 9. 72 743 20 9.80 028 28 O.19 972 9.92 715 8 44
17 9.72763 209.80056
280.19944
9.92707 8 43
18 9.72783 20 9.80084 28 0.19916 9.92699 8 42
19 9.72 803 20 9.80 112 28 O.19 888 9.92 691 8 41
20 9.72 823 20 9.80 140 28 O.19 860 9.92 683 8 40
21 9.72843
9.80168
0.1983~ 9.92675 8 39
22 9.72863
~~9.80
195
~~0.19805 9.92667 8 38
23 9.72 883 19 9.80 223 28 O.19 777 9.92 659 8 37
24 9.72 902 20 9.80 251 28 O.19 749 9.92 651 8 36
25 9.72 922 20 9.80279 28 0.19721 9.92 643 8 35
26 9.72 942
9.80 307 28 0.19 692 9.92 635 8 34
27 9.72962
~~9.80335 28' 0.19665 9.92627 8 33
28 9. 72 982 20 9.80 363 28 O.19 637 9. n 619 8 32
29 9.73002 20 9.80391 28 0.19609 9.92 611 8 31
'30 9.73022
9.80419 28 0.19581 9.92 602 8 30
9.73041
~~9.80447 270.19553
9.92595.8
29
131
32 9 73 061 20 9.80 474 28 O.19 526 9.92 587 8 28
27
133 9:73 081120 9.805021 2810.19498 9.92 579 8
34 9.73101; 20 980530 28'0.19470 9.92571; 81.6
!
9.73 121 19 9.80 558 2810.19 442 9.92 562 1 8 25
135
36 9.73140 209.80586
280.19414
9.92555 9 24
37 9.73 160 20 9.80 614 28 0.19 386 9.92 546 8 23
38 9.73180 209.80642
270.19358
9.92538 8 22
39 9.73200 19 9.80669 28 0.19331 9.92530 8 21
40 9.73 219 20 9.80 697 28 O.19 303 9.92 522 8 20
41 9.73 239 20 9.80 725 28 O.19 275 9.92 514 8 19
42 9.73 259 19 9.80 753 28 0.19 247 9.92 506 8 18
43 9.73278 20 9.80781 27 0.19219 9.92 498 8 17
44 9.73 298 20 9.80 808 28 0.19 192 9.92 490 8 16
45 9.73 318 19 9.80 836 28 O.19 164 9.92 482 9 15
46 9.73337 20 9.80864 28 0.19136 9.92 473 8 14
47 9.73357 209.80892
270.19108
9.92465 8 13
48 9.73377 19 9.80919 28 0.19081 9.92457 8 12
49 9.73396 20 9.80947 28 0.19053 9.92449 8 11
50 9.73 416 19 9.80 975 28 O.19 025 9.92 441 8 10
9
51 9.73435 20 9.81003 27 0.18997 9.92 432 8
8
52 9.73455 199.81030
280.18970
9.92425 9
7
53 9.73474 20 9.81058 28 0.18942 9.92 416 8
6
54 9.73 494 19 9.81 086 27 O.18 914 9.92 408 8
55 9.73 513 20 9.81 113 28 O.18 887 9.92 400 8 5
4
56 9.73 533 19 9.81 141 281O.18 859 9.92 392 8
3
57 9.73 552 20 9.81 169 271O.18 831 9.92 384 8
2
58 9.73572 19 9.81 196 2810.18804 9.92 376 9
1
59 9.73591 20 9.81224 2810.18776 9.92 367 8
60 9.73611
9.81252
10.18748 9.92 3W
0
i
log cos
! d
1470 2370 3210
log cot I cd!
log tan
I
log sin
67°
Id
TABLE III
,
log sin
Prop. Parts
'
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
28
29
27
1 0.5 0.5 0.4
1.0 0.9 0.9
2
3
1.4 1.4 1.4
4
1.9 1.9 1.8
5 2.4 2.3 2.2
6 2.9 2.8 2.7
7 3.4 3.3 3.2
3.9 3.7 3.6
8
9 4.4 4.2 4.0
10 4.8 4.7 4.5
20 9.7 9.3 9.0
30 14.5 14.0 13.5
40 19.3 18.7 18.0
50 24.2 23.3 22.5
19
20
21
1 0.4 0.3 0.3
2 0.7 0.7 0.6
1.0 1.0 1.0
3
1.4 1.3 1.3
4
1.8 1.7 1.6
5
1.9
6 2.1 2.0
7 2.4 2.3 2.2
8 2.8 2.7 2.5
3.2 3.0 2.8
9
10! 3.5 3.3 3.2.
20 I 7 0 6 7 6.
30! 10.5 10.0 9.5
40 114.0 13.3 12.7
50 17.5 16.7 15.8
1
2
3
4
5
6
7
8
9
10
20
30
40
50
9
0.2
0.3
0.4
0.6
0.8
0.9
1.0
1.2
1.4
1.5
3.0
4.5
6.0
7.5
8
0.1
0.3
0.4
0.5
0.7
0.8
0.9
1.1
1.2
1.3
2.7
4.0
5.3
6.7
-
7
0.1
0.2
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.2
2.3
3.5
4.7
5.8
9.73 611
9.73 620
9.73650
9.73 669
9.73 689
9.73 708
9.73 727
9.73747
9.73766
9.73 78~
9.73805
9.73824
9.73 843
9.73 863
9.73 882
9.73901
9. 73 921
9.73940
9.73 959
9.73 978
9.73 997
9.74017
9.74036
9.74 055
9.74074
9.74 093
9.74 113
9.74 132
9.74 151
9.74 170
9.74189
9.74208
9.74 227
9.74246
9.74 265;
9.74 284
9.74303
9.74322
9. 74 341
9.74 360
9.74 379
9. 74 398
9.74417
9.74436
9.74455
9.74474
9.74493
9.74512
9.74531
9.74549
9.74568
9.74587
9.74606
9.74625
9.74644
9. 74 662
9.74 681
9.74700
9.74719
9.74737
9.74756
log cas
Pro". Parts
.-
98
.
Id
330
log tan
19 9.81252
20 9.81279
19 9.81 30Z
20 9.81335
19 9.81 362
19 9.81 390
418
20 9.81
9.81445
19
199.81473
20 9.81500
199.81528
199.81556
20 9.8 J 583
19 9.81 611
19 9.81 638
20 9.81 666
19 9.81 693
19 9.81 721
19 9.81 748
19 9.81 776
20 9.81 803
19 9.81831
19 9.81858
19 9.81 886
19 9.81913
20 9.81941
19 9.81 968
19 9.81 996
19 9.82 023
19 9.82 051
19 9.82078
19 9.82106
19 9.82 133
19 9.82161
19 9.82 188
19 9.82 215
19 9.82243
199.82270
19 9. 82 298
19 9.82 325
19 9.82 352
19 9.82380
19 9.82 40Z
19 9.82435
19 9. 82 462
19 9. 82 489
19 9. 82 517
19 9.82 544
18 9.82571
19 9.82599
19 9.82 626
19 9. 82 653
19
19
18
19
9.82681
9. 82 708
9.82 735
9.82762
19 9.82 790
19 9.82 817
18 9.82844
19 9.82871
9.82 899
d
1460 2360 3260
log
cot
Ic d
log cot
1230
log cas
d
27 0.18748
28 0.18721
28 0.18693
27 0.18665
28 O.18 638
610
28 O.18
58~
27 O.18
0.18555
28
270.18527
28 0.18500
280.18472
270.18444
417
28 O.18
389
27 O.18
28 O.18 362
27 0.18334
28 O.18 307
27 0.18279
28 O.18 252
27 O.18 224
28 O.18 197
27 0.18169
28 0.18142
27 O.18 114
28 0.18087
27 0.18059
28 O.18 032
271O.18 004
28 0.17 977
27 O.17 949
28 O.17 922
27 O.17 894
28, O.17 867
9.92 359
9.92 351
9.92 342
9.92335
9.92 326
9.92 318
9.92 310
9.92 302
9.92293
9.92 285
9.92277
9.92269
9.92 260
9.92 252
9.92 244
9.92235
9.92 227
9.92219
9.92 211
9.92 202
9.92 194
9.92 186
9.92177
9.92 169
9.92 161
9.92 152
9.92 144
9.92 136
9.92 127
9.92 119
9.92 111
9.92 102
9.92 094
27: 0 1i' 81
281O.17 785
270.17757
28 O.17 730
27 0.1770~
27 0.17675
28 0.17 648
27 0.17620
28 0.17593
27 O.17 565
27 O.17 538
28 0.17 511
270.17483
27 0.17456
280.17429
27 0.17401
27 0.17374
28 0.17347
27 0.17319
27 0.17 2n
27 O.17 265
28 O.17 238
270.17210
27 0.17183
27 0.17156
28 0.17129
0.17101
~9.92077 ii
9.92 069
9.92060
9.92 052
9.92044
9.92 035
9.92 027
9.92 018
9.92010
9.92 002
9.91 993
9.91 985
9.91976
9.91968
9.91959
9.91951
9.91 942
9.91 934
9.91925
9.91917
9.91 908
9.91 900
9.91891
9.91883
9.91874
9.91866
9.91 857
8
8
8
9
8
8
8
9
8
8
8
9
8
8
9
8
8
8
9
8
8
9
8
8
9
8
8
9
8
8
9
8
8
271 O. 17 839 9.92 086
Icd
log tan
log sin
56°
9
8
9
8
8
9
8
9
8
8
9
8
9
8
9
8
9
8
9
8
9
8
9
8
9
8
9
d
2130 3030
Prop. Parts
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
I
28
27
0.5 u.4
~] 0.9 0.9
1.4 14
1.9 1.8
2.3 2.2
2.8 2.7
3.3 3.2
3.7 3.6
4.2 4.0
4.7 4.5
9.3 9.0
14.0 13.5
18.7 18.0
23.3 22.5
3
4
5
6
7
8
9
10
20
30
40
50
1
2
3
4
5
6
7
8
9
10
20
30
40
50
19
20
18
0.3 0.3 0.3
0.7 0.6 0.6
1.0 1.0 0.9
1.3 1.3 1.2
1.7 1.6 1.5
2.0
1.9 1.8
2.3 2.2 2.1
2 7 2.5 2.4
3.0 2.8 2.7
3.3 3.2
3.0
6 7 6 3 60
10.0 9.5 9.0
13.3 12.7 12.0
16.7 15.8 15.0
1
2
3
4
5
6
7
8
9
10
20
30
40
50
9
0.2
0.3
0.4
0.6
0.8
0.9
1.0
1.2
1.4
1.5
3.0
4.5
6.0
7.5
8
0.1
0.3
0.4
0.5
0.7
0.8
0.9
1.1
1.2
1.3
2.7
4'.0
5.3
6.7
Prop. Parts
99
340
TABLE ill
log sin
0
1
2
3
4
6
6
7
8
9
10
II
12
13
14
16
16
17
\8
19
20
21
22
23
24
26
26
27
28
29
30
31
32
33
1 log tan ic d I
1d
9.7475Q
9.74775
9.74794
9.74812
9.74 8~1
9.74850
9.74868
9.74887
9.74906
9.74924
9.74 943
9.74961
9.74980
9.74 999
9.75 017
9.75036
9.75 054
9.75073
9. 75 091
9.75110
9.75 128
9.75147
9.75 165
9.75184
9.75202
9.75221
9.75 239
9.75 258
9.75276
9.75294
9.75313
9.75331
9.75350
9.75 368
9.75405
9.75 423
9.75441
9.75459
9.75478
9.75496
9.75514
9.75533
9.75551
44
46
';6
47
48
49
60
51
52
53
54
66
56
57
58
59
60
9.75569
9.75587
9.75605
9.75624
9.75 642
9.75660
9.75 678
9.75696
9.75714
9.75733
9.75751
9.75 769
Q.75 787
9.7580;
9.75823
9.75841
9.75859
1
logcos
0.17101
19 9.82899 27
199.82926
270.17074
189.82953
270.17047
28 0.17020
19 9.82980
9.83
OO~
27 0.16992
19
0.16965
9.83035
18 9.83062 27 0.16938
27
19
199.83089
280.16911
27 0.16883
18 9.83117
19 9.83144 27 0.16856
171 27 O.16 829
18 9.83
\98 27 0.16 80~
19 9.83
27 0.16775
19 9.83225
252 28 O.16 748
18 9.83
19 9.83 280 27 0.16 720
189.83307
270.16693
19 9.83 334 27 O.16 666
189.83361
270.16639
61~
19 9. 83 388 27 O.I6
18 9.83415 27 0.16585
558
19 9.83 442 28 0.16
18 9.83470 27 0.16530
19 9.83497 27 0.16503
18 9.83524 27 0.16476
19 9.83551 27 0.16449
4n
18 9.83578 27 0.16
395
19 9.83 605 27 0.16
18 9.83 632 27 0.16 368
189.83659
2710.16341
19 9.83686 27 0.16 314
18 9.83713 27 0.16287
19 9.83740 28 0.16260
18 9.83 76§ . 27' 0.16232
18?' 8~ ??? 1 271O.16 205
18 9.83 849.
18 9.83876
18 9. 83 ~'\3
19 9.83 9JO
18 9.83957
18 9.83 984
19 9.84011
18 9.84 038
18 9.84 065
18 9.84 092
18 9. 84 119
19 9.84 146
18 9.84173
18 9. 84 200
18 9.84 227
18 9.84254
18 9. 84 280
19 9.84307
18 9.84 334
18 9.84 361
18 9.84 388
18 9.84415
18 9.84442
18 9.84469
18 9. 84 496
9.84 523
--
8
9.91591 9
9.91 582 9
9. 91 ~7~ I 8
9.91 :6), 'J
9.91556 9
9.91547 9
9.91538 8
9.91530 9
9.91521 9
9.91512 8
9. 91 501 9
9.91495 9
9.91 486 9
9.91477 8
9.91469 9
9.91 460 9
9.91451 9
9.91442 9
9.914338
9.91425 9
9.91416 9
9.9\407
9
9.91398 9
9.91 389 8
9.91381 9
9.91372 9
9.9\ 363 9
9.91 354 9
9.9\ 345 9
9.9\ 336
1
dO. 16 151
27 0.16124
27 O. 16 097
27 O. 16 070
27 0.16043
27 0.16 016
27 0.15989
27 O. 15 962
27 O. 15 935
27 0.15 908
27 O. 15 881
27 O. 15 854
27 0.15827
27 O. 15 800
27 O. 15 773
26 0.15746
27 O. 15 720
27 0.15693
27 O. 15 666
27 O. 15 639
27 O. 15 61 ~
27 0.15585
27 0.15558
27 0.15531
27 O. 15 504
0.15 477
2350 3250
60
59
58
57
56
66
54
53
52
51
60
49
48
47
46
46
44
43
42
41
40
39
38
37
36
36
34
33
32
3\
30
29
28
27
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
1
2
3
4
5
6
7
8
9
10
20
30
40
50
28
27
26
0.5
0.4
0.4
0.9
0.9
0.9
1.4
1.4
1.3
1.9
1.8
1.7
2.3
2.2
2.2
2.8
2.7
2.6
3.3
3.2
3.0
3.7
3.6
3.5
4.2
4.0
3.9
4.7
4.5
4.3
9. 3
9. 0
8.7
14.0 13. 5 13 .0
18.718.017.3
23.3 22.5 21.7
1
2
3
4
5
6
7
8
9
10
')n.
30
40
50
I
1
\
19
0.3
0.6
1.0
1.3
1.6
1.9
2.2
2.5
2.8
3.2
h
1
9.5
12.7
15.8
9
1 0.2
2 0.3
3 0.4
4 0.6
5 0.8
6 0.9
7 1.0
8 1.2
91.41.2
10 1.5
20 3. 0
304.54.0
40 6. 0
50 7.5
18
0.3
0.6
0.9
1.2
1.5
1.8
2.1
2.4
2.7
3.0
ihJl
9.0
12.0
15.0
55°
d
log tan
cd I log cot
h
8
0.1
0.3
0.4
0.5
0.7
0.8
0.9
1.1
1.3
2. 7
5.3
6.7
100
1260
log cos
0 9.75859 189.845p
270.15477
9.91336
1 9.75877 189.84550
260.15450
9.91328 8
2 9.75895 18 9.84576 27 0.15424 9.91319 9
3 9.75913 18 9.84603 27 0.15397 9.91310 9
4 9.75931 18
9.84630 27 0.15370 9.9130\ 99
5 9.75949 18 9.84657 27 0.15343 9.91292
6 9.75967 18 9.84684 27 0.15316 9.91283 99
7 9.75985 189.84711
270.152899.91274
8 9.76003 189.84738
'60.15262
9.91266 89
9 9.76021 18 9.84764
0.15236
9.9\ 257
9
10 9.76039 18 9.84791 27 0.15209 9.91248 9
11 9.76 05Z 18 9.8481§ 27 0.15 182 9.91 239 9
12 9.76075 18 9.84845 27 0.15155 9.91230 9
13 9.76093 18 9.84872 27 0.15128 9.91221 9
\4 9.76111 18 9.84899 26 0.15 10~ 9.91 212 9
15 9.76129 17 9.84925 27 0.15075 9.91203 9
16 9.76146 18 9.84952 27 0.15048 9.91 194 9
17 9.76164 18 9.84979 27 0.15021 9.91185 9
18 9.76182 18 9.85006 27 0.14994 9.91 176 9
19 9.76200 189.85033
260.14967
9.91167 9
20 9.76218 18 9.85059 27 0.14941 9.91 158 9
21 9.76236 17 9.85086 27 0.14914 9.91 149 8
22 9.76253 18 9.85113 27 0.14887 9.91141 9
23 9.76271 18 9.85140 26 0.14860 9.91 132 9
24 9.76289 18 9.85166 27 0.14834 9.91 123 9
25 9.76307 17 9.85193 27 0.14807 9.91111 9
26 9.76324 18 9.85220 27 0.14780 9.91 105 9
27 9.76342 189.85247
260.14753
9.91096 9
28 9.76360 189.85273
270.14727
9.91087 9
29 9.76378 179.85300
270.14700
9.91078 9
30 9.76395 18 9.85327 27 0.14673 9.91069 9
31 9.76413 18 9.85354 26 0.14646 9.91060 9
32 9.76431 17 9.85380 27 0.14620 9.91051 9
339.76448189854072701093991042927
34 9.7646611Q 9.85434!7"lo.14566
9.9103311n
359.76484117
9.85460i 17'0.145409.9102319
36 9.76501 18 9.85487 27 0.14513 9.91014 9
37 9.76519 18 9.85514 26 0.14486 9.91005 9
38 9.76537 179.85540
270.14460
9.90996 9
39 9.76554 189.85567
270.14433
9.90987 9
40 9.76572 18 9.85594 26 0.14406 9.90978 9
41 9.76 590 17 9.85 620 27 O.14 380 9.90 969 9
42 9.7660Z 18 9.85647 27 0.14353 9.90960 9
43 9.76625 17 9.85674 26 0.14326 9.90951 9
44 9.76642 18 9.85700 27 0.14300 9.90942 9
46 9.76 660 17 9.85 727 27 0.14 273 9.90 933 9
46 9.76677 189.85754
260.14246
9.90921 9
47 9.76695 17 9.85780 27 0.14220 9.90915 9
48 9.76712 18 9.85807 27 0.14193 9.90906 10
49 9.76730 17 9.85834 26 0.14166 9.90896 9
50 9.7674Z 18 9.85860 27 0.14140 9.90887 9
51 9.76 765 17 9.85 887 26 0.14 1\3 9.90 878 9
52 9.76782 189.85913
270.14087
9.90869 9
53 9.76 800 17 9.85 940 27 O.14 060 9.90 860 9
54 9.76 817 18 9.85 967 26 O.14 033 9.90 851 9
55 9.76835 17 9.85993 27 0.14007 9.90842 10
56 9.76 852 18 9.86 020 26 O.13 980 9.90 832 9
57 9.76 870 17 9.86 046 27 0.13 954 9.90 823 9
58 9.76887 17 9.86073 27 0.13927 9.90811 9
59 9.76 904 18 9.86 100 26 0.13 900 9.90 805 9
60 9.76922
9.86126
0.13874 9.90796
I log cos ! d I logcot led! logtan I logsin I d I
1
':uts
Oil co
1460
8
9
8
9
8
9
8
9
8
9
9
8
9
8
9
9
8
9
8
9
9
8
9
9
8
9
9
8
9
9
350
TABLE III
log sin
Prop. Parts
Idl
9.91857
9. 91 849
9.91 840
9.91832
9.91 82~
9.91815
9. 91 806
9.91798
9.91789
9.91781
9.91772
9.91 76~
9.91755
9.91746
9.91 738
9.91729
9 . 91 720
9.91712
9.91 70~
9.91695
9. 91 686
9.91 677
9.91 669
9.91660
9.91 651
9.91643
9.91 634
9.91625
9.91617
9.91 608
9.91599
[
1
36
36
37
38
39
40
41
42
43
log cot
1240 2140 3040
2150 3050
Prop. Parts
60
59
58
57
56
65
54
53
52
27
1 1 0.4
2 ' 0.9
3
4
5
6
7
8
9
10
20
30
40
50
51
50
49
48
47
46
46
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
26
26
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
6
4
3
2
1
0
'
I
26
0.4
0.9
1.4
1.3
1.8
1.7
2.2
2.2
2.7
2.6
3.2
3.0
3.6
3.5
4.0
3.9
4.5
4.3
9 .0
8. 7
13. 5 13.0
18. 0 17. 3
22.5 21.7
9.0
12.0
15.0
17
0.3
0.6
0.8
1.1
1.4
1.7
2.0
2.3
2.6
2.8
5.7
II.)
11.3
14.2
9
8
1 0.2 0.1
2 0.3 0.3
3 0.4 0.4
4 0.6 0.5
5 0.8 0.7
6 0.9 0.8
7 1.0 0.9
8 1.2 1.1
91.41.2
10 1.5 1.3
20 3. 0 2.7
30 4. 5 4.0
40 6.05.3
50 7.5 6.7
I
I
Prop. Parts
10
0.2
0.3
0.5
0.7
0.8
1.0 ,.
1.2
1.3
1.5
1.7
3.3
5:"U
6.7
8.3
l
TABLE III
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
2\
22
23
24
25
26
27
28
29
30
31
32
9.77 524
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
9.77 541
9.77558
9.77575
9.77592
9.77 609
9. 77 626
9.77643
9.77 660
9.77 677
9. 77 694
9.77711
9.77 728
9.77744
9.77761
9.77 778
9.77795
9.77812
9.77 829
9. 77 846
9. 77 862
9.77 879
9.77 896
9.77913
9.77 930
1159
60 9.77 946
-
1
log cos
1430 2330
17dl
17 9.86126
18 9.86153
179
17 9.86
17 9.86 206
189.86232
17 9.86259
285
17 9.86
18 9.86312
17 9.86 338
179.86365
392
17 9.86
41
18 9.86
17 9.86445
179.86471
17 9.86 498
18 9.86 524
9.76922
9.76939
9.76957
9.76974
9.76991
9. 77 009
9.77 026
9.77043
9.77061
9.77078
9. 77 095
9.77 112
9. 77 130
9.77147
9.77 164
9.77 181
9.77 199
9.77 216
9.77233
9.77 250
9. 77 268
9.77285
9. 77 302
9.77 319
9.77336
9.77 353
9.77 370
9.77387
9.77 405
9.77 422
9.77439
9.77456
9.77 473
34 9. 77 507
35
36"
I d I log tan
logsin
17 9.86 551
17 9.86 577
17 9.86 603
18 9.86 630
17 9.86 656
17 9.86 683
17 9.86 709
17 9.86 736
179.86762
17 9.86 789
17 9.86 8\5
18 9.86 842
17 9.86868
17 9.86 894
17 9.86921
17 9. 86 947
17 9.86 974
1
I
log cot
i logcosIdl
874
27 0.13
847
26 0.13
821
27 O.13
794
26 O.13
27 0.13768
26 0.13741
715
27 O.13
26 0.13688
27 O.13 66~
270.13635
26 O.13 608
~27O.13 582
26 0.13 555
270.13529
26 O.13 502
270.13476
26 O.13 449
26 0.13423
27 O.13 397
26 0.13370
27 O.13 344
26 O.13 317
27 O.13 291
26 O.13 264
27 0.13238
26 O.13 211
27 0-.13 185
26 O.13 158
26 0.13 132
27 O.13 106
26 O.13 079
270.13053
26 O.13 026
9.90796
9.90787
9.90 777
9.90 768
9.907~9
9.90750
9.90 741
9.90731
9.90 722
9.90713
9.90 704
9.90 694
9.90685
9.90676
9.90 667
9.90657
9. 90 648
9.90639
9.90 630
9.90620
9.90 61\
9.90 602
9.90 592
9.90 583
9.9057~
9.90 565
9.90 555
9.90 546
9.90537
9.90 527
9.90 518
9.90509
9.90 499
9
10
9
9
9
9
10
9
9
9
10
9
9
9
10
9
9
9
10
9
9
10
9
9
9
10
9
9
10
9
9
10
9
17 9. 87 027 I. 26I O.12 973 9. 90 480 I '9
260.1297 4 9.90471 9
17 9.87053
17 9.87 079 27 O.12 921 9.90 462 10
17 9.87106 26 0.12894 9.90452 9
17 9.87132260.128689.904439
17 9.87158 27 0.12842 9.90434 10
17 9.87185 26 0.12 815 ~.90421 9
17 9.87 211 27 O.12 789 9.90 415 10
9.90405 9
17 9.87238 260.12762
17 9.87 264 26 0.12 736 9.90 396 10
17 9.87 290 27 O.12 710 9.90 386 9
17 9.873\7 26 0.12683 9.90377 9
17 9.87343 26 0.12657 9.90 368 10
16 9.87369 27 0.12631 9.90 358 9
17 9.87396 26 0.12604 9. 90 349 10
17 9.87422 26 0.12578 9.90 339 9
17 9. 87 448 27 0.12552 9.90330 10
17 9.87475 26 O.12 525 9.90 320 9
17 9.87501 26 0.12499 9.90311 10
17 9.87 527 27 0.12473 9.90301 9
16 9.87554 26 0.12446 9.90 292 10
90 282 9
17 9.87580 26 0.12420 9.
10
17 9.87606 27 0.12394 9.90273
17 9.87633 26 0.12367 9.90 263 9
0.12341
9.
90
254
10
17 9.87659 26
16 9.87685 26 0.12315 9.90 244 9
0.12289 9.90235
9.87711
I d 1 log cot
3230
cd!
log tan
~--- - ----
--
log sin
53°
d
I
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
3\
30
29
28
1
2
3
4
5
6
7
8
9
26
20,
25
30
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
I' I
TABLEm
i26° 2160306.
Prop.Parts
,
40
50
I
17
0.3
0.6
0.8
1.1
1.4
1.7
2.0
2.3
2.6
2.8
5 7
17
17
17
16
17
17
16
17
17
16
17
17
16
17
17
16
17
16
17
17
16
17
16
17
16
17
16
17
16
17
16
17
16
~. ~~478
494 ,
60
9.0 8.5
12.0 11.3
15.0 14.2
1
2
3
4
5
6
7
8
9
10
20
30
40
50
10
0.2
0.3
0.5
0.7
0.8
1.0
1.2
1.3
1.5
1.7
3.3
5.0
6.7
8.3
Prop.
16
0.3
0.5
0.8
1.1
1.3
1.6
1.9
2.1
2.4
2.7
5,
8.0
I
oiiiii-.
10.7
13.3
9
0.2
0.3
0.4
0.6
0.8
0.9
1.0
1.2
1.4
1.5
3.0
4.5
6.0
7.5
Parts
102
..
--
37"
Id
0 9.77946
I 9.77963
2 9.77 980
3 9.77997
4 9.78013
5 9.78 030
6 9.78047
7 9.78 063
8 9.78080
9 9. 78 097
10 9.78 113
11 9.78 130
12 9.78147
13 9.78 J63
14 9.78180
15 9.78197
16 9.78213
17 9.78 230
18 9.78 246
19 9.78 263
20 9.78 280
21 9.78 296
22 9.78313
23 9.78 329
24 9.78 346
25 9.78 362
26 9.78379
27 9.78395
28 9.78412
29 9.78428
30 9.78445
31 9.7846\
27
26
1 0.4 0.4
2 0.9 0.9
1.4 1.3
3
4
1.8 1.7
5
2.2 2.2
6 2.7 2.6
3.2 3.0
7
8
3.6 3.5
3.9
9 4.0
10 4.5 4.3
20 9.0 8.7
30 13.5 13.0
40 18.0 17.3
50 22.5 21. 7
18
0.3
0.6
0.9
1.2
1.5
1.8
2.1
2.4
2.7
log sin
log tsn
9.87711
9.87738
9.87764
9.87790
9.87817
9.87843
9.87869
9.87895
9.87922
9.87948
9.87974
9.88000
9.88027
9.88053
9.88079
9.88105
9.88 131
9. 88 158
9. 88 184
9.88210
9.88236
9. 88 262
9.88289
9.88315
9. 88 341
9.88367
9.88393
9.88420
9. 88 446
9.88472
9. 88 498
9.88524
9.88550
Q 88 577
9 88 603
cd
27
26
26
27
26
26
26
27
26
26
26
27
26
26
26
26
27
26
26
26
26
27
26
26
26
26
27
26
26
26
26
26
27
log cot
O. 12 289
0.12262
0.12236
0.12210
0.12 183
0.12 157
0.12 131
0.12 105
0.12078
O. 12 052
O. 12 026
0.12000
0.11 973
0.11 947
0.11 921
0.11 895
0.11 869
0.11 842
0.11 816
0.11 790
0.11 764
0.11 738
0.11 711
0.11 685
0.11 659
0.11 633
0.11 607
0.11 580
0.11 554
0.11 528
0.11 502
0.11 476
0.11 450
~0.11 423
H
34 9. 78 510,
,
26 O.11 397
35 9.78 527 I 16 9. 88 62~ I 26! O. 11 371
36 9.78543
17 9.88655260.11345
37 9.78 560 16 9.88681
26 0.11 319
38 9.78576
16 9.88 707 26 O. 11 293
39 9.78 592 17 9.88 733 26 O. 11 267
40 9.78 609 16 9.88759
27 0.11 241
41 9.78 625 17 9.88786
26 0.11 214
42 9.78642
16 9. 88 812 26 0.11 188
43 9.78 658 16 9.88838
26 0.11 162
44 9.78 674 17 9. 88 864 26 0.11 136
45 9.78691
26 0.11 110
16 9.88890
46 9.78707
16 9. 88 916 26 0.11 084
47 9.78723
16 9. 88 942 26 0.11 058
48 9.78739
17 9. 88 968 26 0.11 032
49 9.78756
16 9. 88 994 26 0.11 006
50 9. 78 772 16 9.89020
26 0.10980
51 9.78 788 17 9.89046
27 O. 10 954
52 9.78805
16 9.89073
26 0.10927
53 9.78821
16 9.89099
26 0.10901
54 9.78837
16 9.89125
26 0.10875
55 9.78853
9.89
151
16
26 O. 10 849
56 9.78 869 17 9.89 177 26 0.10823
57 9.78886
16 9.89203
26 0.10797
58 9.78 902 16 9.89229
26 0.10771
59 9.78918
16 9.89255
26 0.10745
0.10719
9.89281
60 9.78 934
log cos
log cot! cd! log tan
d
i7
I I
142"
2320 3220
log cos
1270 2170 3070
Id
9.90235 10
9.90 225
9
9.90216 10
9. 90 206 9
9.90 197 10
9.90 \87 9
9.90 178 10
9.90168
9
9.90 159 10
9. 90 149 10
9 . 90 139 9
9. 90 130 10
9. 90 120 9
9.90111 10
9.90101 10
9.90091
9
9. 90 082 10
9.90 072 9
9. 90 063 10
9.90053 10
9.90043
9
9.90 034 10
9.90 024 10
9.90014
9
9.90005 10
9.89 995 10
9.89985
9
9.89976 10
9.89966 10
9.89 956 9
9.89947 10
9.89 937 10
9.89927
9.89918 ,~
9 89 908 10
9.89 898 !10
9.89 888 9
9.89879 10
9.89 869 10
9.89 859 10
9.89849
9
9.89840 10
9.89830 10
9.89 820 10
9.89 810 9
9.89801 10
9.89791 10
9.89 781 10
9.89 771 10
9.89761
9
9.89752 10
9.89742 10
9.89732 10
9.89722 10
9.89712 10
9.89 702 9
9.89 693 10
9.89
9.89 68'[IQ
673 10
9.89663 10
9.89653
I log sin I d
52°
Prop. Parts
I
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
1
2
3
4
5
6
7
8
9
10
20
30
40
50
27
26
0.4
0.4
0.9
0.9
1.4
1.3
1.8
1.7
2.2
2.2
2.7
2.6
3.2
3.0
3.6
3.5
4.0
3.9
4.5
4.3
9.0
8.7
13.5 13.0
18.0 17.3
22 . 5 21.7
17
0.3
1
0.6
2
0.8
3
1.1
4
1.4
5
6
1.7
2.0
7
8
2.3
9
2.6
10
2.8
20
5 7
30 I 8.5
40 [II. 3
50 \4.2
1
2
3
4
5
6
7
8
9
10
20
30
40
50
I
10
0.2
0.3
0.5
0.7
0.8
1.0
1.2
1.3
1.5
1.7
3.3
5.0
6.7
8.3
Prop.
16
0.3
0.5
0.8
1.1
1.3
1.6
1.9
2.1
2.4
2.7
5 3
8.0
10.7
13.3
9
0.2
0.3
0.4
0.6
0.8
0.9
1.0
1.2
1.4
1.5
3.0
4.5
6.0
7.5
Parts
103
TABLE
log
0
sin
9. 78 934
9.78 950
I
2 9.78967
3
9.78 983
4
5
9.78999
log tan
d
16
17
16
16
16
log cos
log cot
cd
9.89281
9.89 307
9 89 333
9.89 359
9.89 385
9.89653
9.89 643
9.89 633
9.89 624
9.89 614
2610.10719
O. 10 693
26
O. 10 667
26
O. 10 641
26
0.10 615
26
O. 10 589
26
9.79 095
16
9.89541
II 9.79 111 17 9.89567
9.89 593
12 9. 79 128
16
9.89619
13 9. 79 144
16
9.89645
14 9. 79 160
16
15 9.79176 16 9.89 671
16 9.79192 16 9.89 697
17
9. 79 208
18
9. 79 224
9.79240
9.79 256
9.79272
19
20
21
22 9.79288
23
9.79304
24
9. 79 319
25 9.79335
9.89 723
16
9.89749
16
9.79 383
9.79 399
9. 79 415
9.79431
9.79447
33 9.79463
)4 9.79478
9.79 494
'35
36
9. 79 510
37
9.79 526
38
39
40
41
42
43
44
45
46
9.79542
9.79558
9.79573
\47
9. 79 684
9.79 589
9.79605
9. 79 621
9.79 636
9.79 652
9. 79 668
26
26
26
26
26
16
26
9.89905
16 9.89931
16
16
16
16
9.90002
9.90 035
9.90061
9.90086
10
10
9
10
10
169.90475
15
9. 90
501
16 9.90 527
149
150
9.79715
169.90553
159.90578
'51
'52
9.79 746
9. 79 762
53 9.79778
54 9.79 793
55
9. 79 809
16 9. 90 604
16 9.90 630
15 9. 90 656
16
9. 90
682
16 9.90 708
15 9.90 734
56 9.79825
57 9. 79 840 169.90759
58 9.79 856 16 9.90 785
59 9.79872 15 1).90811
60
26
26
25
26
10
10
9.89 534
9.89524
9.89514
10
10
10
9.89
9. 89
9. 89
9.89
9
10
10
10
501
49~
48~
47~
0.10225
0.10199
0.10173
0.10147
0.10121
9.8946~
9.89 45~
9.89 44~
9.89 43~
9.89 42~
10
10
10
10
10
0.10095
9.89415
10
26 0.10069
0.10043
26
26
9.89554
9.89544
9.89 40~
'10
9.89 39~ 10
0.10017
9.89 38~
10
0.09991
0.09 965
0.09939
0.09914
9.89375
9.89 364
9.89354
9.89344
11
10
10
10
9.79887
9.'10837
26 0.09 525
26 0.09499
'
0
1
25
26
0.4 0.4
54
53
52
2
3
4
0.9
1.3
1.7
0.8
1.2
1.1
51
5
2.2
2.1
50
49
6
7
2.6
3.0
2.5
2.9
48
8
3.5 3.3
47
46
45
44
43
9
10
20
30
40
50
42
2
3
4
39
38
37
34
33
32
31
3:>
3.8
4.2
8.3
13.0
12.5
16
1
0.3 0.3
2
0.6
3
0.8 0.8
4
5
1.1
1.1
1.4
1.3
6
7
1.7
2.0
1.6
1.9
2.3
2.6
2.8
5 7
2.1
2.4
2.7
5.3
50
8.5
11.3
8.0
lJ.7
7.5
10.0
0.5
9.89 112 11
6
9.89101
9.89 091
26 0.09241
26 0.09215
26 0.09 189
0.09 163
9.89081 10
9.89071 11
9.89 060
110
9.89 050
5
4
3
12
11
7
8
9
10
20
30
40
50
10
9
8
7
r
I
11
10
9
0.2
0.2
0.2
61°
9.89050
10
60
16
9.90914
16
9.90 940
15
9.90966
9.91069
25
0.09 086
26
0.09 060
26
9.89 020
11
9.89 009
10
0.09034 9.88999
0.09008 9.88989 1
~~I
10.08982 9.88978
0.08957 9.88968 1~
~~i
9.88958
2610.08931
10
57
56
55
~54
53
52
51
9.80 166
9.80 182
9.80197
1~
9.91 250
9.91276
15. 9.91 301
~~0.08750 9.88 886 1~
0.08724 9.88875
25 O.08 699 9.88 865 10
0.08673
~~I
0.08647
9.91 327
:~
9.91353
16
9.88 855:~
9.88844
10
26
25
0.4
0.9
1.3
0.4
0.8
1.2
5
1.7
2.2
1.7
2.1
6
2.6
2.5
7
3.0 2.9
1
2
3
4
8
9
10
20
3.5
3.9
4.3
8.7
3.3
3.8
4.2
8.3
44
30
13.0
12.5
43
40
50
17.3 16.7
42
41
40
0.7
0.6
1.8 1.7 1.5
3.7
5.5
7.3
9.2
3.3
5.0
6.7
8.3
9.80 290
9.80305
9.80320
9.80336
9.80 351
9.80 366
3.0
4.5
6.0
7.5
35
36
9.80428'
9.80443
37
38
39
40
41
9.80458
9.80473
9.80489
9.80504
9.80519
42
43
9.80 5~4
9.8055Q
46
9.80 595
47
48
49
60
51
52
53
50t
9.80 610
9.80625
9.80 641
9.80 656
9.80671
9.80 686
9.80701
9.80 716
55 9.80731
56 9.80746
'.80 761
9.80 777
9.80792
9.80 807
57
58
59
60
I
104
15
15
16
15
15
16
9.91 507
9.91533
9.91 55~
9.91585
9.91 610
9.91 636
10
9.88772
11
9.88761
10
9.88751
10
9.88 741
11
9.88 730
10
26
0.08467
26'
0.08441
26
0.08415
25
0.08 390
26
0.08 364
26
9.91 732 I 26! 0.08261
15
0.08235
9.91765
26
15
9.91791
15 9.91816
16
9.91842
15
9.91868
15
9.91893
15
9.91912
16
9.91945
15
0.08209
0.08184
0.08158
0.08 132
0.08 107
25
26
26
25
26
008081
26
0.08055
26
33
32
31
30
29
28
27
26
21.7
20.8
_.
.
I
log
cos
15
15
16
15
15
15
15
15
15
9.92 048
9.92073
9.92 09~
9.92 125
9.92 150
9.92 176
9.92 2:12
9.92 227
26
0.07 952
0.07927
0.07 901
0.07 875
0.07850
0.07 824
0.07798
0.07 773
25
26
26
25
26
26
25
26
Id
1400 2300 32C"
log cot
I
log
tan
2
3
0.5
0.8
0.5
0.8
4
1.1
1.3
1.6
1.9
2.1
1.0
1.2
1.5
1.8
2.0
2.4
2.7
5.3
2.2
2.5
5.0
8.0
7.5
10.0
5
6
7
8
9
10
20
30 i
i
23
11 22
10 21
11
20
10
19
11 18
50
9.88 61~
10
9.88605
11
11
10
17
1
2
3
0.2
0.2
4
0.7
5
6
0.9 0.8
9.88668
9.88657
9.88647
9.886%
9.88626
9.88 563
9.88552
9.88 542
9.88 531
9.88521
9.88 510
9.88499
9.88 489
10
11
10
11
10
11
11
10
11
26
0.07 670 9.88 447 110
11
26
0.07 644 9.88436 11
25\
0.07 619 9.88 425
cd
16
15
0.3 0.2
25
24
0.07747 9.88478 10
9.92253
15 9.92279 26
2510.07721 9.88468 11
16
0.07 696 '.88 457
'.92 304
15
9.92 330
15
9.92 356
15
9.92 381
1
9 88688 !
10
9.88678
10
44 9.80565 15 9.91971 25 0.08029 9.88594 10 16
45 9.80580 15 9.91996 26 0.08004 9.88584 11 15
14
0.07 978 9.88 573
9.92 022
0.4 0.3 0.3
0.6 0.5 0.4
Pro. Parts
1610 23103210
2610.09163
0.09137 9.89040
59
15 9.90863 26 0.09 111 9.89 030 10 58
9.90 889
10
32 9.80382, 15 9.91662 26 0.08338 9.88720 11
33 9.80397I
9.91688
0.08312 9.88709
34 9.80412i :~ 9 91713 I n: 0.08 287 9.88 699:~
14.2 13.3 12.5
0.9 0.8 0.8
1.1 1.0 0.9
1.3 1.2 1.0
1.5 1.3 1.2
1.6 1.5 1.4
26 0.09292
25 0.09 266
9.80 136
26
27
28
29
30
31
5
6
26 0.09 318
9.90837
Prop. Parts
26
21 9.80213 15 9.91379 25,0.08621 9.88834 10 39
0.08596 9.88824
38
22 9.80228
9.914J4
0.08570 9.88813:h
37
23 9.80244 :~ 9.91430
~~!
24 9.80259 15 9.91456 2610.08544 9.88803 10 36
25 9.80274 16 9.91482 25 0.08518 9.88793 11 35
34
0.08 493 9.88 782
13
10
d
16
18
19
20
0.7
9.89 122
log cos
50
49
48
47
46
45
21.7
20.8
log cot
cd
10 9.80043
9.91095
0.08905 9.88948
11 9.80058
9.91 121
~~0.088799.88937:h
1~
12 9.80074
9.91147
,0.08853 9.88927
13 9.80082 1~ 9.91 172
0.08828 9.88917 1~
~~I
0.08802 9.88906 10
14 9.80105
15 9.91198 26
15 9.80120
9.91224
0.08776 9.88896
17 9.80151
4
26 0.09 344
16
tan
~9.90
992
7 9.79996 1 9.91018
8 9.80012
1~ 9.91043
9 9.80027
17.3 16.7
14
11
10
10
10
10
lolt
'
35
9.89173
9.89162
9.89152
9.89142
9.89132
d
16
3.9
4.3
8.7
17
36
26 0.09473
25 0.09447
26 0.09422
26 0.09396
26 0.09370
9.79 918
9.79 9~4
9.79 950
5 9.79965
6 9.79981
41
4:>
10
10
9.79887
1290 2190 309°
89°
III
sin
1 9.79903
56
55
9.89 193
9.89183
10
10
log
60
59
58
57
29
8
28
9
0.09 888 9.89 334 110
26
16 9.90 112
10
27
15 9.90 138 2610.09862 9.89324 il0 ...\-)
"'1" 20
2610.09836
9.89314110
16 9.90164
30
() 25
0.09
810
9.89
3041
9.90
190
26
16
40
24
0.09784
9.89294
9.90216
10
26
16
50
23
26 0.09758 9.89284 10
16 9.90242
22
0.09732
9.89274
9.90268
10
26
16
21
15 9.90294 26 0.09706 9.89264 10
20
9.89 254 10
26 0.09 680
16 9.90 320
25 0.09654 9.89244 11 19
16 9.90 346
169.90371
26 0.09629 9.89233 10 13
17
0.09603
9.89223
10
26
1
15 9.90 397
16
169.90423
26 0.09577 9.89213 10
2
15
9.89203
3
10
26 0.09551
16 9. 90 442
\48 9 79 699
9. 79 731
O. 10 407
26
0.10381
26
0.10355
26
O. 10 329
26
0.10 303
26
0.10 277
26
0.10251
26
169.89775
16 9.89801
16 9.89827
16 9.89853
15 9.89872
26 9.79351 16 9.89957
9.89983
27 9. 79 367
16
28
29
30
31
32
0.10459
26
0.10433
26
Prop. Parts
d
9.89 604 10
9.79015 16 9.89 411
6 9.79031 16 9.89437 26 0.10563 9.89594 10
7 9.79047 16 9.89463 26 0.10537 9.89584 10
8 9. 79 063 16 9.89489 26 0.10511 9.89574 10
9 9.79079 16 9.89 515 26 0.10 485 9.89 564 10
10
TABLE
1230 218~ 3080
88°
ill
I
log
60°
sb
13
12
11
10
9
8
7
6
5
4
40 110.7
13.3 12.5
0.4 0.3
0.6 0.5
1.1
1.3
1.5
1.6
7
8
9
0.7
1.0
1.2
1.3
1.5
10
1.8 1.7
20
30
3.7
5.5
40
50
7.3 6.7
9.2 8.3
3.3
5.0
3
2
1
0
I d I 'I
Prop. Parts
105
l
40°
log
sin
d
0 9.80 807
1 9.80 822
2 9.80837
3 9.80 852
4 9.80 867
5 9.80 882
6 9.80 897
7 9.80 912
8 9.80927
9 9.80 942
10 9.80 957
11 9.80 972
12 9.80 987
13 9.81 002
14 9.81 017
15 9.81032
16 9.81 047
17 9.81 061
18 9.81076
19 9.81 091
20 9.81 106
21 9.81 12 t
22 9.81 136
23 9.81 151
24 9.81 166
25 9.81 180
26 9.81 195
27 9.8121Q
28 9.81225
29 9.81 240
30 9.81 254
31 9.81 269
32 9.81 284
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
1390
9.81314:
9.81328'
9.81343
9.81358
9.81 372
9.81 387
9.81402
9.81 417
9.81431
9.81 446
9.81 461
9.81475
9.81 49Q
9.81505
9.81 519
9.81 534
9.81 549
9.81 563
9.81578
9.81 592
9.81 607
9.81 622
9.81 636
9.81651
9.81 665
9.81680
9.81694
2290
log
tan
9.92 381
15
9.92 407
15
159.92433
15 9.92 458
15 9.92 484
15 9.92 510
15 9.92 535
15 9.92 561
9.92 587
15
15 9.92 612
15 9.92 638
15 9.92 663
15 9.92 682
15 9.92 715
15 9.92 740
15 9.92 766
14 9.92792
15 9.92 817
15 9.92 843
15 9.92 868
15 9.92 894
15 9.92 920
15 9.92 945
15 9.92 971
14 9.92 996
15 9.93 022
15 9.93 048
15 9.93073
15 9.93099
14 9.93 1~4
15 9.93 150
15 9.93 175
I cd I
log
cot
IOgCOSldl
26 0.07 619
0.07 593
26
250.07567
26 0.07 542
26 0.07 516
25 0.07 49Q
26 0.07 465
26 0.07 439
25 0.07413
26 0.07 388
25 0.07 362
26 0.07 337
26 0.07 311
25 0.07 285
26 0.07 260
26 0.07234
25 0.07208
26 0.07 183
25 0.07157
26 0.07 132
26 0.07 106
25 0.07 08Q
26 0.07 055
0.07 029
25
26 0.07 004
0.06 978
26 0.06 952
25 0.06927
26
25,0.06901
0.06 876
26
0.06 85Q
25 0.06 825
26' 0.06 799
9.88 42~ 10
9.88 415
9.88 404 II
10
9.88394
11
9.88 383
II
9.88372
10
9.88 362 11
9.88351
11
9.88 340 10
9. 88 330 11
9 . 88 319 11
9.88 308
9.88 298 10
11
9.88 287 11
9.88276
10
9.88 26§ 11
9.88255
11
9.88244
10
9.882341142
9.88 223
11
9 . 88 212
9.88 20 I 11
9. 88 191 10
11
9.88180
9. 88 I 69 11
11
9.88 158
10
9 . 88 148
11
9.88137
9.88 126 11
11
9.88 II
10
9.88105
9.88 094 11
9.93
201
9.88
083 111
,11
I 26!
15 .
.
9.88
071
I
Ii,
0.06748
9 88061
14 9 93252
26
itO
0.06722
9.88 051
15 9.93278
25
9.93303
2610.06697
9.88 040 11
15
11
0.06671
9.88029
14 9.93329
25
11
15 9.93 354 26 0.06 646 9.88018
11
15 9.93 380 26 0.06 620 9.88 007 11
9.93406
0.06594
9.87
996
15
25
11
14 9.93 431 26 0.06 569 9. 87 98~ 10
9.93457
006543
9.87975
15
25
11
9.93482
0.06518
9.87964
15
26
11
9.93
508
0.06
492
9.87953
14
25
II
0.06467
9.87942
15 9.93533
26
11
9.93 559
15
25 0.06 441 9.87931
11
0.06416
9.87920
14 9.93584
11
9.93 610 26 0.06 390 9.87909
15
26
11
9.93
636
0.06
364
9.87
898
15
25
11
9.93 661
0.06
339
9.87887
14
9.93 687 26 0.06 313 9.87 877 10
15 9.93712
25
9.87 86§ 11
14
26 0.06288
9.93 738
0.06 262 9.8785511 11
15 9.93763
25
9.87 844
15
26 0.06237
11
9.93 789
0.06 211 9.8783311
14 9.93 814 25 0.06 186
822
15 9.93840
26 0.06 16Q 9.87
9.87 811 11
14 9.93 865 25 0.06 135 9.87 800 11
26 0.06109
15 9.93891
9.87 789 11
25
14
11
9.87 778
9.93916
0.06084
3190
1
1
49°
TABLE III
1300 2200 3100
~
log sin
Prop. Parts
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
41
40
39
38
37
36
35
34
33
32
31
30
29
28
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
0
4r
I d I log tan
cd I
I
log cos
d
26 0.06 058 9.87 767
1 9.81709 15
25 0.06033 9.87756
14 9.93942
9.93
967
2 9.81 723 15
26 0.06007 9.87745
9.93993
3 9.81738 14
'25
1
2
3
4
5
6
7
8
9
10
20
30
40
50
I
1
2
3
4
5
6
7
8
9
20:
30
40
50
I
I
4
5
6
7
8
26
25
0.4
0.4
0.9
0.8
1.3
1.2
1.7
1.7
2.2
2.1
2.6
2.5
3.0
2.9
3.5
3.3
3.9
3.8
4.3
4.2
8.7
8.3
13 . 0 12. 5
17.3 16.7
21 . 7 20. 8
15
0.2
0.5
0.8
1.0
1.2
1.5
1.8
2.0
2.2
14
0.2
0.5
0.7
0.9
1.2
1.4
1.6
1.9
2.1
5.0
7.5
10. 0
12. 5
4.7
7.0
9. 3
11.7
9.81 752
9.81767
9. 81 781
9. 81 796
9. 81 81Q
9.81
9.81
9.81
9.81
9.81
839
854
868
882
897
15 9.81 911
-
.~
11
10
1 0.2 0.2
2 0.4 0.3
3 0.6 0.5
4 0.7 0.7
5 0.9 0.8
6 1.1 1.0
7 1.3 1.2
8 1.5 1.3
9 1.6 1.5
10 1.8 1.7
20 3.7 3.3
30 5.55.0
407.36.7
50 9. 2 8. 3
16
17
18
19
20
21
22
23
9.81 926
9. 81 94Q
9.81955
9. 81 969
9.81983
9.81 998
9. 82 012
9.82 026
24
9.82 04
25
'26
27
28
29
30
31
32
9.82 055
9.82069
9.82 084
9.82 098
9.82 112
9.82 126
9.82 141
9.82 155
35
36
37
38
39
40
41
42
43
44
45
46
9.82198 i
9.82 212
9.82 226
9.82 24Q
9.82255
9.82 269
9.82 283
9.82297
9.82311
9.82 326
9.82 340
9.82354
48
49
50
51
52
53
54
55
~
-1
60'
.
14
15
14
14
14
15
14
14
9.94 554
9.94579
9.94 604
9.94 630
9.94 655
9.94 681
9.94 706
9.94732
2610.05 \92
25 0.05 166
25 0.05 141
26 0.05 116
25 0.05090
26 0.05 065
25 0.05 039
26 0.05014
25 0.04988
25 0.04 963
26 0.04 938
25 0.04912
9.87390
9.87 378
9.87 367
9.87 35§
9.87345
9.87 334
9.87 322
9.87311
9.87300
9.87 288
9.87 277
9.87262
9.82 382
9.82 396
9.82410
9.82424
9.82 439
9.82 453
9.82 467
25
26
25
25
26
25
26
0.04 861
0.04 836
0.0481Q
0.04785
0.04 760
0.04 734
0.04 709
9.82 48 t
14 9.95 317
9.82509
9.82 523
9.82537
9.82551
14 9.95368
14 9.95 393
14 9.95418
9.95444
25
26
25
25
26
0.04 683
0.04658
0.04632
0.04 607
0.04582
57
58
59
,60
I
log cos
lJ!S6
228'
14 9.95342
Id I
313'
log cot
11
11
11
11
11
11
11
11
11
11
11
11
11
12
11
11
11
11
11
11
11
11
12
11
11
11
11
11
12
11
11
11
26 0.05982 9.87734
956 9.87 723
25 0.05
931 9.87 712
26 0.05
905 9.87 701
25 0.05
880 9.87 690
26 0.05
25 0.05 854 9.87 679
829 9.87 668
26 0.05
9.87657
25 0.05803
778 9.87 64§
26 0.05
9.87635
25 0.05752
26 0.05 727 9.87 624
9.87613
25 0.05701
9.87601
26 0.05676
65Q 9.87590
25 0.05
26 0.05 625 9.87 579
25 0.05 599 9.87 568
26 0.05574 9.87557
25 0.05 548 9.87 54§
26 0.05 523 9.87 535
497 9.87 524
25 0.05
26 0.05472 9.87513
25 0.05446 9.87501
25 0.05 421 9.87 490
26 0.05 396 9.87 479
25 0.05 37Q 9.87468
26 0.05345 9.87457
25 0.05319 9.87446
26 0.05294 9.87434
25 0.05 268 9.87 423
26!00~?1~ ?~!~!~!
14 9.94808 i
14 9.94 834
14 9.94 859
884
15 9.94
14 9.}4 910
14 9.94 935
14 9.94 961
14 9.94986
15 9.95012
14 9.95 037
14 9.95 062
14 9.95088
14 9.95113
14 9.95 139
14 9.95 164
14 9.95190
15 9.95215
14 9.95 240
14 9.95 266
14 9.95 291
56 9.82495
,..
14 9.94 528
33 9.82169 I 15 9.94757
47 9.82368
106
9.940\8
9.94044
9. 94 062
9. 94 095
9. 94 120
14 9.94146
171
15 9.94
14 9.94197
14 9.94 222
248
15 9.94
14 9.94 273
15 9.94299
14 9. 94 3~4
94 350
15 9.
14 9.94375
14 9. 94 40I
15 9.94426
14 9.94 452
14 9. 94 477
15 9.94 503
15
14
15
14
15
9 9.81 825
10
11
12
13
14
Prop.
Part.s
0.06 084 9.87 778 11
59
11
9. 93 916
9. 81 694
log cot
1310 22103110
1
TABLE III
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
1
2
3
4
5
6
7
8
9
10
20
30
40
50
11
11
11
11
12
11
11
12
11
11
11
23
22
21
20
19
18
17
16
15
14
9.87 243
9.87 232
9.87221
9.87209
9.87 198
9.87 187
9.87 175
11
11
12
11
11
12
11
12
11
10
9
8
7
6
9.87 164
9.87153
9.87141
9.87 130
9.87119
11
12
11
11
12
5
4
3
2
1
26 0.04887 9.87255 12 13
Ic d I
0.04556
log tan
I
9.87107
log sin
I
15
0.2
0.5
0.8
1.0
1.2
1.5
1.8
2.0
2.2
2.5
.i.Jl
I
7 5
30
40 I 10.0
50 [ 12.5
1
2
3
4
5
6
7
8
9
10
i 12 25
24
26
25
0.4
0.4
0.9
0.8
1.3
1.2
1.7
1.7
2.2
2.1
2.6
2.5
3.0
2.9
3.5
3.3
3.9
3.8
4.3
4.2
8. 7 8 . 3
13. 0 12. 5
17.316.7
21 . 7 20. 8
1
2
3
4
5
6
7
8
9
10
20
30
40
50
14
0.2
0.5
0.7
0.9
1.2
1.4
1.6
1.9
2.1
2.3
li
7 0
9.3
11.7
12
11
0.2 0.2
0.4 0.4
0.6 0.6
0.8 0.7
1.0 0.9
1.2 1. 1
1.4 1.3
1.6 1.5
1.8 1.6
2.0 1.8
4. 0 3 . 7
6. 0 5. 5
8.07.3
10'.0 9.2
0
d
Prop. Parts
l
42°
TABLE III
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
1
135
36
37
38
39
40
41
42
43
44
45
46
;47
48
,49
50
51
52
53
54
56
56
57
58
59
60
,
9.82 551
9.82 565
9.82579
9.82 593
9.82607
9.82621
9.82 635
9.82 649
Ic d I 106 cot I
tan
1320
log cas
0.04 556
0.04 531
0.04505
0.04 48Q
0.04455
0.04429
0.04 404
9.87 107
9. 87 09~
9.87085
9.87 073
9.87062
9.87050
9.87 039
0.04 378
0.04353
14 9.95647
25
14 9.95 672 26 0.04 328
9.95 698 25, 0.04 302
14 9.95
723 25 0.04 277
14
748 26 0.04 252
14 9.95
9.95774
25 0.04226
14
14 9.95 79~ 26 0.04201
25 0.04 1!5
14 9.95825
13 9.95850 25 0.04 15~
14 9.95875 26 0.04125
14 9.95 901 25 0.04 099
14 9.95926 26 0.04074
14 9.95 952 25 0.04 048
14 9.95 977 25 0.04 023
14 9.96 002 26 0.03 998
13 9.96 028 25 0.03 972
14 9.96 053 25 0.03 947
14 9.96 078 26 0.03 922
14 9.96 104 25 0.03 896
14 9.96 122 26 0.03 871
14 9.96155 25 0.03845
13 9.96180 25 0.03820
14 9.96 205 26 0.03 795
14 9.96231 25 0.03769
14 9.96256 25 0.03744
9.83010
13 9.96281 ! 26 0.03719
9.83 023 I 14 9.96 307 1 251 0.03 693
2510.0:' oM
9.83037 14 9.96 m
9.8305!
14 9.96357 26 0.03643
9.83 065 13 9.96 383 25 0.03 617
9.83 078 14 9.96 408 25 0.03 592
9.83 092 14 9.96 433 26 0.03 567
9.83 106 14 9.96 459 25 0.03 541
9.83 120 13 9.96 484 26 0.03 516
9.83 133 14 9.96 51Q 25 0.03 490
9.83147 14 9.96535 25 0.03465
9.83 161 J3 9 96 560 2610.03 440
9.83 174 14 9.96 586 25i 0.03 414
9.83188 14 9.96611 25' 0.03389
9.83202 13 9.96636 26 0.03364
9.83 215 14 9.96 662 25 0.03 338
9.83229 13 9.96687 25 0.03313
9:83242 14 9.96712 26 0.03288
9.83 256 14 9.96 738 25 0.03 262
9.83270 13 9.96763 25 0.03237
9.83 283 14 9.96 788 26 0.03 212
9.83 297 13 9.96 814 25 0.03 186
9.83310 14 9.96839 25 0.03161
9.83 324 14 9.96 864 26 0.03 136
9.83338 13 9.96890 25 0.03110
9.83 351 14 9.96 915 25 0.03 085
9.83 365 13 9.96 940 26 0.03 060
9.83378
9.96966
0.03034
9.87 028
9.8701§
9.87 005
9.86 993
9.86 982
9.86 970
9.86959
9.86947
9.86936
9.86924
9.86913
9.86 902
9.86890
9.86 879
9.86 867
9.86 855
9.86 844
9.86 832
9.86 821
9.86 809
9.86 798
9.86 78~
9.86775
9.86 763
9.86752.
9.86740
14
14
14
14
14
14
14
14
9.82663
9.82 677
9.82 691
9.82 705
9.82 719
9.82733
9.82747
9.8276!
9.82775
9.82788
9.82 802
9.82816
9.82 830
9.82 844
9.82 858
9.82 872
9.82 885
9.82 899
9.82 913
9.82 927
9.8294!
9.82955
9.82 968
9.82982
9.82996
9.95 444
9.95 462
9.95495
9.95 520
9.95545
9.95571
9.95 596
9.95 622
25
26
25
25
26
25
26
25
1
1
0
1
2
3
4
5
6
7
T<il- log
1
1
'I
logsin
d
log cot
Ic d I log tan
I
9.86728
9.86
717
d
I
1
9.86705 i
9.86694
9.86 682
9.86 670
9.86 659
9.86 647
9.86 635
9.86 624
9.86612
9.86 600
9.86 589
9.86577
9.86565
9.86 554
9.86542
9.86530
9.86 518
9.86507
9.86 495
9.86 483
9.86472
9.86 460
9.86448
9.86 43§
9.86 425
9.86413
log sin
11
II
12
11
12
11
11
12
11
12
11
12
11
12
11
12
11
11
12
11
12
12
11
12
11
12
11
12
11
12
11
12
12
11
12
111
12
12
11
12
12
11
12
12
11
12
12
11
12
12
12
11
12
12
11
12
12
12
11
12
d
2220 3120
60
59
58
57
56
66
54
53
52
51
5~
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
6
4
3
2
1
0
14
1
0.2
2
0.5
3
0.7
4
0.9
5
1.2
6
1.4
7
1.6
8
1.9
9
2.1
10
2.3
20 I 4.7
30 I 7.0
40 ' 9.3
50 . 11.7
[
1
2
3
4
5
6
7
8
9
10
20
30
40
50
log sin
26
26
0.4
0.4
0.9
0.8
1.3
1.2
1.7
1.7
2.2
2.1
2.6
2.5
3.0
2.9
3.5
3.3
3.9
3.8
4.3
4.2
8.7
8.3
13. 0 12. 5
17. 3 16. 7
21 .7 20.8
1
2
3
4
5
6
7
8
9
10
20
30
40
50
13
0.2
0.4
0.6
0.9
1.1
1.3
1.5
1.7
2.0
2.2
4.3
~
8.7
10.8
--.
~
-
12
11
0.2 0.2
0.4 0.4
0.6 0.6
0.8 0.7
1.0 0.9
1.2 1.1
1.4 1.3
1.6 U
1.81.6
2.0 1.8
4.0 3.7
6. 0 5 . 5
8.0 7.3
10 .0 9. 2
Prop. Parts
430
TABLE III
Prop. Parts
.
0
1
2
3
4
6
6
7
8
9
10
11
12
13
14
16
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
d
log tan
Ic d I
log cot
9.83378
O.03 034
9. 96 966
251 0.03009
9.83 392 14 9.96991
13
25
9.83405
9.97 016 26 0.02 984
9. 83 419 14
139.97042250.02958
9.83432
0.02933
14 9.97067
25
9.83446 13 9.97 092 26 0.02 908
9.83 459 14 9.97118
0.02882
25
9.83473 13 9.97143
25 0.02857
9.83 486 149.97168
250.02832
9.83 500 13 9.97193
26 0.02807
9.83 513 14 9.97 219 25. 0.02 781
9.83 527 13 9.97 244
0.02 756
251 0.02 731
9.83540 14 9.97 262 26
9.83 554 13 9.97 295 25 0.02 705
9.83 567 14 9.97 320 25 O.02 68~
9.83581 13 9.97345
26 0.02655
9.83 594 149.97371250.02629
9.83 608 13 9.97 396 25 0.02 604
9.83621 13 9.97421
260.02579
9.83 634 14 9.97447
25 0.02553
9.83 648 139.97472
250.02528
9.83661 13 9.97 497 26 0.02 503
9.83 674 14 9.97523
0.02477
9.83 688 13 9.97 548 25
25 0.02 452
9.83 701 14 9.97 573 25 0.02 427
9.83715 13 9.97 598 26 0.02 402
9.83 728 13 9.97 624 25 0.02 376
9.83741 14 9.97649
25 0.02351
9.83755 13 9.97 674 26 0.02 326
9.83768 13 9.97 700 25 0.02 300
9.83 781 14 9.97 725 25 0.02 2!5
9.83795 13 9.97 750 26 0.02 250
9.83808 13 9.97 776 251 0.02 ?24
9.83821 I 13 9.97 801 1 251 0.02 199
1
tog cas
1360
!
9.86 020
.
25 0.02 171
26 0.02 149 9.85 996
9.97826
13 9.97 851
13 9.97 877 25 0.02 123
13 9.97 902 25 0.02 098
14 9.97 927 26 0.02 073
13 9.97 953 25 0.02 047
13 9.97 978 25 0.02 022
13 9.98 003 26 0.01 997
149.98029250.01971
13 9.98 054 25 0.01 946
13 9.98 079 25 0.01 921
13 9.98104 26 0.01896
13 9.98130 25 0.01 87Q
14 9.98155 25 0.01845
13 9.98 180 26 0.01 820
13 9. 98 206 25 O.01 794
13 9.98231 25 0.01769
13 9.98256 25 0.01744
13 9.98281 2610.01 719
13 9.98 307 25' 0.01 693
14 9.98332 2510.01668
13 9.98357 26 0.01643
13 9.98 383 25 0.01 617
13 9.98 408 25 0.01 592
13 9.98433 25 0.01567
139.98458
260.01542
9.98484
0.01 516
1
d
2260 3160
log cot
led
I
log tan
9.85 984
9.85 972
9.85 960
9.85 948
9.85 936
9.85 924
9.85912
9 . 85 900
9.85 888
9.85876
9. 85 864
9. 85 851
9. 85 839
9.85 827
9.85 815
9.85 803
9. 85 791
9.85 779
9.85 766
9.85754
9.85742
9.85 730
9.85718
9. 85 706
9.85693
log sin
46°
Prop. Parts
d
9. 86 413
9.86 40I
9.86 389
9.86377
9.86 366
9. 86 354
9.86342
9.86330
9.86 318
9. 86 30~
9.86 295
9.86 283
9.86271
9. 86 259
9. 86 247
9.86 235
9.86 223
9.86211
9. 86 200
9. 86 188
9.86176
9.86 164
9.86152
9.86140
9.86 128
9. 86 116
9.86 104
9.86092
9.86 080
9. 86 068
9.86 056
9.86 044
9.86 032
1
9.83831
35 9.83848
36 9.83 861
37 9.83874
38 9.83887
39 9. 83 901
40 9.83914
41 9.83 927
42 9.83940
43 9.83954
44 9.83 967
46 9.83 980
46 9.83 993
47 9. 84 006
48 9.84 020
49 9.84 033
50 9.84 046
51 9.84 059
52 9.84 072
53 9.84085
54 9.84 098
55 9. 84 112
56 9.84 125
57 9.84138
58 9.84 151
59 9.84164
60 9.84177
1330 2230 313°
log cas
12
12
12
11
12
12
12
12
12
11
12
12
12
12
12
12
12
11
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
1
60
59
58
57
56
65
~4
53
52
51
50
49
48
47
46
46
44
43
42
41
40
39
38
37
36
36
34
33
32
31
30
29
28
12 27
..
12
12
12
12
12
12
12
12
12
12
12
13
12
12
12
12
12
12
13
12
12
12
12
12
13
d
25
24
23
22
21
20
19
18
17
16
16
14
13
12
11
10
9
8
7
6
6
4
3
2
1
0
26
26
1
0.4
0.4
2
0.9
0.8
3
1.3
1.2
4
1.7
1.7
5
2.2
2.1
6
2.6
2.5
7
3.0
2.9
8
3.5
3.3
9
3.9
3.8
10
4.3
4.2
20
8. 7 8. 3
3013.012.5
4017.316.7
50 21.7 20.8
14
0.2
0.:5
0.7
0.9
1.2
1.4
1.6
1.9
2.1
1.7
7.0
9.3
11.7
13
0.2
0.4
0.6
0.9
1.1
1.3
1.5
1.7
2.0
2.2
H
65
8.7
10.8
1
2
3
4
5
6
7
12
0.2
0.4
0.6
0.8
1.0
1.2
1.4
11
0.2
0.4
0.6
0.7
0.9
1. 1
1.3
8
1.6
U
1
2
3
4
5
6
7
8
9
10
20
30
40
50
9
10
20
30
40
50
I
I
2.3
1.81.6
2.0 1.8
4.0 3. 7
6 .0 5 . 5
8.0 7.3
10. 0 9. 2
Prop. Parts
109
l
440
TABLE III
log sin
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
\6
17
18
19
20
21
22
23
d
9.84 177
9. 84 190
9.84 203
9. 84 216
9.84229
9.84242
9.84255
9.84269
9. 84 282
9.84295
9. 84 308
9.84 321
9.84334
9 .84 347
9.84360
9.84373
9.84 385
9.84 398
9.84411
9.84424
9.84437
9.84 450
9.84 463
9.84476
13
13
13
13
13
13
14
13
13
13
13
13
13
13
13
12
13
13
13
13
13
13
13
13
log tan
cd
9. 98 484
9. 98 509
9. 98 534
9. 98 560
9.98 585
9. 98 610
log cot
log eos
0.0 I 516 9.85 693
0.01491 9. 85 681
0.01466 9. 85 669
25 0.01 440 9.85657
25 0.01 415 9. 85 645
0.01 39Q 9.85632
25 0.01
365
25
25
26
9.98635
9. 98 661
9.98 686
9.98 711
9.98737
9. 98 762
9.98787
9. 98 812
9.98 838
9.98 863
9. 98 888
9.98913
9.98 939
9. 98 964
9.98 989
9.99015
9. 99 040
9.99 065
9.85 620
9.85 608
9.85 596
9.85 583
9.85571
9.85 559
9.85547
9.85 534
9.85522
9.85 510
9.85 497
9 . 85 485
9.85473
9.85 460
9.85 448
9.85436
9.85423
9.85411
26
25 0.01 339
25 0.01 314
26 O.01 289
25 0.01 263
25 O.0 I 238
25 0.01 213
26 0.01 188
25 0.01 162
2510.01 137
2510.01 1\2
26 0.0 I 087
25 0.01 061
25 0.01036
26 0.01 011
25 0.00 985
25 0.0096Q
25 O.00 935
24 9.84 489 13 9. 99 090 26 0.00910
25 9.84 502 13 9.99116 25 0.00884
26 9. 84 515 13 9.99141 25 0.00859
27 9. 84 528
28
9. 84 540
29 9.84553
30 9.84 566
31 9. 84 579
32 ~.
.
12 9. 99 166
13 9.99191
25 0.00834
26 O.00 809
13 9.99217 25 0.00783
13 9.99242 25! 0.00758
13 9.99 267 26: 0.00733
~. ~~~?~
25' 2. 22 72~
~~~~~13
.
v.vv
,~
Z5
'
.34 9.84. 618 13
j 25,0.00657
12 9.99343
.
35 9.84 630 13 9.99 368
0.00 632
136 9.84643 13 9. 99 394 261
25 O.00 606
'37
38
39
40
9.84
9. 84
9. 84
9. 84
656
669
682
694
41
42
43
44
9.84707
9. 84 720
9.84 733
9.84745
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
9.84758
9.84771
9.84784
9. 84 796
9.84809
9.84 822
9.84 835
9. 84 847
9. 84 860
9.84873
9.84885
9.84898
9.84911
9.84923
9.84936
9.84949
:=J
log cos
13
13
12
13
9. 99 419
9.99 444
9. 99 46~
9.99495
25 O.00 581 9.85 237
25 0.00 556 9.85225
26 0.00 531 9. 85 212
9.85200
25 0.00505
9.99 520
9.99545
9.99 570
9.99596
25
25
26
25
0.00480
0.00455
0.00430
0.00404
9.85 187
9. 85 175
9.85 162
9.85150
13
13
12
13
13
13
12
13
13
12
13
13
12
13
13
9. 99 621
9.99 646
9.99672
9.99 697
9. 99 722
9.99747
9.99773
9.99 798
9.99 823
9.99 848
9.99874
9.99 899
9. 99 924
9.99949
9.99 975
0.00000
25
26
25
25
25
26
25
25
25
26
25
25
25
26
25
0.00379
0.00354
0.00328
0.00303
0.00 278
0.00253
0.00227
0.00202
0.00177
0.00152
0.00126
0.00101
0.00076
0.00051
0 00 025
0.00000
log tan
9.85137
9.85 125
9.85112
9.85 100
9.85 087
9.85074
9 . 85 062
9. 85 049
9.85 037
9.85 024
9.85012
9.84999
9. 84 986
9.84974
9. 84 96\
9.84949
1S5° 2250 3150
log cot
EJ
J log sin
45°
224° 314°
Prop. Parts
I
12 60
59
12 58
12 57
12 56
13
I
12 55
I
12
12
13
12
12
12
13
12
12
13
12
12
13
12
12
13
12
12
12
13
12
13
12
13
12
13
12
13
12
13
13
12
13
12
13
12
13
13
12
13
12
23
"q2
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
I
:11,.
'""
I
.,
1J
541
53
52
51
50
49
48
47
46
45
44
43
42
4\
40
39
38
37
9.85 399 13 36
9.85 386 12 35
9.85374 13
9.85361
33
12 341
9.85 349 12 32
9.85337 13 31
9.85324
30
9.85 312 12
13 29
28
~.~~
~~~12
. VJ ..v,
13
9.85274 12 26
9. 85 262 12 25
9.85250 13 24
13
13
12
13
CD
134°
d
I
1
2
3
4
5
6
7
8
9
10
20
30
40
50
26
0.4
0.9
1.3
1.7
2.2
2.6
3.0
3.5
3.9
4.3
8.7
13.0
17.3
25
0.4
0.8
1.2
1.7
2.1
2.5
2.9
3.3
3.8
4.2
8.3
12.5
16.7
21.7
20.8
&.
I
TABLE
I
NATURAL
-~Y,
I
14
13
12
1
2
3
4
0.2
0.5
0.7
0.9
0.2
0.4
0.6
0.9
0.2
0.4
0.6
0.8
5
6
7
8
1.2
1.4
1.6
1.9
1.1
1.3
1.5
1.7
1.0
1.2
1.4
1.6
9
10
20
30
40
50
2.1
2.3
4.7
7.0
9.3
11.7
2.0
2.2
4.3
6.5
8.7
10.8
1.8
2.0
4.0
6.0
8.0
10.0
..~ .
'..
If
'iI<!
,l
~, .
'~t'
'j
,'"
I
~Pmp. Parts
110
.
IV
TRIGONOMETRIC
FUNCTIONS
Of angles for each minute from 0° to 90°, correct
to five significant figures
(For explanation, see page 29.)
l
0°
90° 180° 270°
,
I
0
1
2
3
4
6
6
7
8
9
10
11
12
\3
14
16
16
17
18
19
20
21
22
23
24
26
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
46
46
47
48
49
60
51
52
53
54
66
56
57
58
59
60
sin
tan
.00000
.00000
I
00
029
029
058
058
087
087
116
116
.00145 .00145
175
175
204
204
233
233
262
262
.00291 .00291
320
320
349
349
378
378
407
407
.00436 .00436
465
465
495
495
524
524
553
553
.00582 .00582
611
611
640
640
669
669
698
698
.00727 .00727
756
756
785
785
814
815
844
844
.00873 .00873
902
902
931
931
960
960
.00989 1.00989
.01018 ;.01018
047
047
076
076
105
105
134
135
.01164 .01164
193
193
222
222
251
251
280
280
.01309 .01309
338
338
367
367
396
396
425
425
.01454 .01455
483
484
513
513
542
542
571
571
.01600 .01600
629
629
658
658
687
687
716
716
.01745 .01746
I
I
CDS
cot
179' 269' 359'
TABLE IV
cot
3437.7
1718.9
1145.9
859.44
687.55
572.96
491. 11
429.72
381.97
343.77
312.52
286.48
264.44
245.55
229.18
2\4.86
202.22
190.98
180.93
171.89
163.70
156.26
149.47
143.24
137.51
132.22
127.32
122.77
118.54
114.59
110.89
107.43
104.17
p01.11
:98.218
95.489
92.908
90.463
88.144
85.940
83.844
81.847
79.943
78.126
76.390
74.729
73.139
71.615
70.153
68.750
67.402
66.105
64.858
63.657
62.499
61.383
60.306
59.266
58.261
57 290
I
89°
tan
cos
I
1.0000
000
000
000
000
1. 0000
000
000
000
000
1.0000
.99999
999
999
999
.99999
999
999
999
998
.99998
998
998
998
998
.99997
997
997
997
996
.99996
996
996
995
99~
[.99995
995
994
994
994
.99993
993
993
992
992
.99991
991
991
990
990
.99989
989
989
988
988
.99987
987
986
986
985
.99985
60
59
58
57
56
66
54
53
52
51
60
49
48
47
46
46
44
43
42
41
40
39
38
37
36
36
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16'
16
14
13
12
11
10
9
8
7
6
6
4
3
2
1
0
sin
'
1°
91° 181°2710
920182°272°
sin
0 .03490 .03492
1
519
521
2
548
550
3
577'
579
4
606
609
6 .03635 .03638
6
664
667
7
693
696
sin I tan I cot I cos I
0 .01745 .01746 57.290 .99985 60
1
774
775 56.351
984 59
2
803
804 55.442
984 58
3
832
833 54.561
983 57
4
862
862 53. 709
983 56
6 .01891 .01891 52.882 .99982 66
6
920
920 52.081
982 54
7
949
949 51.303
981 53
8 .01978 .01978 50.549
980 52
9 .02007 .02007 49.816
980 51
10 .02036 .02036 49. 104 .99979 60
11
065
066 48.412
979 49
12
094
095 47.740
978 48
13
123
124 47.085
977 47
14
152
153 46.449
977 46
15 .02181 .02182 45.829 .99976 46
16
211
211 45.226
976 44
17
240
240 44.639
975 43
18
269
269 44.066
974 42
19
298
298 43.508
974 41
20 .02327 .02328 42.964 .99973 40
21
356
357 42.433
972 39
22
385
386 41.916
972 38
23
414
415 41.411
971 37
24
443
444 40.917
970 36
25 .02472 .02473 40.436 .99969 36
26
501
502 39.965
969 34
27
530
531 39.5~6
968 33
28
560
560 39.057
967 32
29
589
589 38.618
966 31
3~ .02618 .02619 38. 188 .99966 30
31
647
648 37.769
965 29
32
676
677 37.358
964 28
33
705
706 136.956
963 27
34
734
35
36
37
38
39
40
41
42
43
44
46
46
47
48
49
60
51
52
53
54
55
56
57
58
59
60
.02763
792
821
850
879
.02908
938
967
.02996
.03025
.03054
083
112
141
170
.03199
228
257
286
316
.03345
374
403
432
461
.03490
cos
m
735 136.563
.02764
793
822
85\
\ 881
.02910
939
968
.02997
.03026
.03055
084
114
143
172
.03201
230
259
288
317
.03346
376
405
434
463
.03492
I
cot
QQfl
I
,.99962 26
961 24
960 23
959 22
959 21
.99958 20
957 19
956 18
955 17
954 16
.99953 16
952 14
952 13
951 12
950 11
.9994910
948 9
947 8
946 7
945 6
.99944 5
943 4
942 3
941 2
940
I
.99939 0
I
I
lP70a
8
"'=-
sin
,"':'fl6
723
1
TABLE IV
,
cos
28.636 .99939 60
.399
938 59
28.166
937 58
27.937
936 57
.712
935 56
27.490 .99934 65
.271
933 54
27.057
932 53
725 26.845
931
52
9
752
754
.637
930 51
10 .03781 .03783 26.432 .99929 60
11
810
812
.230
92749
12
839
842 26.031
926 48
13
868
871 25.835
925 47
14
897
900
.642
924 46
15 .03926 .03929 25.452 .99923 45
16
955
958
.264
922 44
17.03984.0398725.080
921 43
18 .04013 .04016 24.898
919 42
19
042
046
.719
918 4\
20 .04071 .04075 24.542 .99917 40
21
100
104
.368
916 39
22
129
133 : 196
915 38
23
159
162 24.026
913 37
24
188
191 23.859
912 36
26 .04217 04220 23.695 .99911 35
26
246
250
.532
910 34
27
275
279
.372
909 33
28
304
308
.214
907 32
29
333
337 23.058
906 31
30 .04362 .0436622.904.99905
30
31
391
395
.752
904 29
32
420
424
.602
902 28
33
449
454
.454
901 27
963
,36.178
35.801
35.431
35 070
34.7\5
34.368
34.027
33.694
33.366
33.045
32.730
32.421
32.118
31.821
31.528
31.242
30.960
30.683
30.412
30. 145
29.882
29.624
29.371
29. 122
28.877
28.636
tan
1
2°
34
478
483 I .308
35 .04507 1.04512 i22. 164
36
536
541
37
565
570
38
594
599
39
623
628
40 .04653 .04658
41
682
687
42
711
716
43
740
745
44
769
774
45 .04798 .04803
46
827
833
47
856
862
48
885
891
49
914
920
60 .04943 .04949
51 .04972 .04978
52 .05001 .05007
53
030
037
54
059
066
66 .05088 .05095
56
117
124
57
146
153
58
175
182
59
205
212
60 .05234 .05241
cos
"H:::OEl
II
I
cot
1770 ;:670 3570
I
900
26
I 99898 25
20.206 .99878 10
20.087
876
19.970
875
.855
873
.740
872
19.627 .99870
.516
869
.405
867
.296
866
. 188
864
19.081 .99863
I
tan
8T
I
sin
tan
cot
cos
O
I
2
3
4
6
6
7
8
9
10
11
12
13
14
16
16
17
18
19
20
21
22
23
24
25
26
27
28
29
3J
31
32
33
34
.05234
263
292
32\
350
.05379
408
437
466
495
.05524
553
582
611
640
.05669
698
727
756
785
.05814
844
873
902
931
.05960
.05989
.06018
047
076
.06105
134
163
192
.05241
270
299
328
357
.05387
416
445
474
503
.05533
562
591
620
649
.05678
703
737
766
795
.05824
854
883
912
941
.05970
.05999
.06029
058
087
.06116
145
175
204
19.081
18.976
.871
.768
.666
18.564
.464
.366
.268
.171
18.075
17.980
.886
.793
.702
17.611
.521
.431
.343
.256
17.169
17.084
16.999
.915
.832
16.750
.668
.587
.507
.428
16.350
.272
.195
. 119
.99863
861
860
858
857
.99855
854
852
851
849
.99847
846
844
842
841
.99839
838
836
834
833
.99831
829
827
826
824
. 99822
821
819
817
815
.99813
812
810
808
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
60
51
52
53
54
65
56
57
58
59
I
.06250..06262
221
233116.043
291
321
350
379
.06408
438
467
496
525
.06554
584
613
642
671
.06700
730
759
788
817
.06847
876
905
934
963
61) .06976 .06993
22.022
897 24
21. 881
896 23
.743
894 22
.606
893 21
21. 470 .9981)2 20
.337
890 19
.205
889 18
21.075
888 17
20. 946
886 16
20.819 .99885 16
.693
883 14
.569
882 13
.446
881 12
.325
879 11
9
8
7
6
6
4
3
2
1
0
sin
279
308
337
366
.06395
424
453
482
511
.06540
569
598
627
656
.06685
714
743
773
802
.06831
860
889
918
947
COB
113
93° 1830 273°
3°
I
cot
86'"
I
60
59
58
57
56
66
54
53
52
51
60
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29.
28
27
26
26
803 24
801 23
799 22
797 21
.99795 20
793 19
792 18
790 17
788 16
.99786 16
784 14
782 13
780 12
778 11
.99776 10
774 9
772 8
770 7
768 6
.99766 6
764 4
762 3
760 2
758 1
.99756 0
I sin I '
806
.15.969 1.99804
.895'
.821
.748
.676
15.605
.534
.464
.394
.325
15.257
.189
.122
15.056
14.990
14.924
.860
.795
.732
.669
14.606
.544
.482
.421
.361
14.301
I
tan
l'jijV
~OOV ilbijV
~
40
94° 184° 274°
,
0
I
2
3
4
5
6
7
8
9
10
1\
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
I
sin
II
tan
.06976 .06993
.07005 .07022
034
051
063
080
092
110
.07121 .07139
150
168
179
197
227
208
237
256
.07266 .07285
295
314
324
344
353
373
382
402
.07411 .07431
440
461
469
490
498
519
527
548
.07556 .07578
585
607
614
636
643
665
672
695
.07701 .07724
730
753
759
782
788
812
817
841
.07846 .07870
875
899
904
929
933
958
962.07987
.07991 .08017
.08020
046
049
075
078
104
107
134
.08136 .08163
192
165
194
221
223
251
252
280
.08281 .08309
310
339
339
368
368
397
397
427
.08426 .08456
455
485
484
5\4
513
544
542
573
.08571 .08602
600
632
629
661
658
690
720
687
.08716 .08749
I
cos
I
cot
1750 2650 3550
I
TABLE
cot
14.301
.241
.182
.124
.065
14.008
13.951
.894
.838
.782
13.727
.672
.617
.563
.510
13.457
.404
.352
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.248
13.197
.146
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13.046
12.996
12.947
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.850
.801
.754
12.706
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12.47-1
.429!
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12.251
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12.035
11. 992
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11.826
.785
.745
.705
.664
11. 625
.585
.546
.507
.468
11. 430
I tan
85°
cos
I
sin
0
1
2
3
4
'5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
.99756 60
754 59
752 58
750 57
748 56
.99746 55
744 54
742 53
740 52
738 51
.99736 50
734 49
731 48
729 47
727 46
.99725 45
723 44
721 43
719 42
716 41
.99714 40
712 39
710 38
708 37
705 36
.99703 35
701 34
699 33
696 32
694 31
.99692 30
689 29
687 28
685 27
683
26
I.99680
25
678 24
676 23
673 22
671 21
.99668 20
666 19
664 18
661 17
659 16
.99657 15
654 14
652 13
649 12
647 11
.99644 10
642
9
639
8
637
7
635
6
.99632
5
630
4
627
3
625
2
622
1
.99619
0
I
sin
50
IV
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
I
tan]
95° 185° 276°
cot
I
cos
.O871§ .08749
745
778
774
80~
803
837
831
866
.08860 .08895
889
925
918
954
947.08983
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.09005 .09042
034
071
063
101
092
130
121
159
.09150 .09189
179
218
208
247
237
277
266
306
11.430 .99619
.392
617
.354
614
.316
612
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609
11.242 .99607
.205
604
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602
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599
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11.059 .99594
11.024
591
10.988
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10.883 .99580
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578
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575
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572
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570
.09295 .09335 10.712 .99567
324
365
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564
353
394
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562
382
423
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559
411
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556
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469
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570
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545
556
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542
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614
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I
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50 .10164 .10216 9.7882 .99482
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246 .7601
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221
275.7322
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53
250
305 .7044
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54
279
334.6768
470
55 . 10308 . 10363 9.6493 .99467
56
337
393.6220
464
57
366
422.5949
461
458
58
395
452.5679
40
8
114
cos
I
cot
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tan
i
sin
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7
6
5
59
424
481 .5411
455
60 .10453 .10510 9.5144 .99452
I '
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
II
10
9
I
6°
sin
I
,
60;
59
58
57
56
56
54
53
52
51
50
49
48
47
46
45
44
43"42
41
1
35
96° 186° 276°
'
.174-0 26403540
".
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0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
16
16
17
18
19
20
21
22
23
24
26
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
5\
52
53
54
55
56
57
58
59
60
----.!.-
.10453
482
511
540
569
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626
655
684
713
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771
800
829
858
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060
089
118
147
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205
234
263
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349
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523
552
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638
667
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783
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840
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100
129
158
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1
coS
tan
I
TABLE
cot
cos
I
. i0510
540
569
599
628
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687
716
746
775
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834
863
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040
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128
158
187
217
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276
305
335
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570
600
629
659
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718
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.12013
042
072
101
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160
190
219
249
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9.5144 .99452
!
!
cot
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.4878
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9.3831
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9.2553
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8.9860
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8.8919
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8.7769
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8.6648
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8.5555
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8.4490
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8.3450
.3245
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.2838
.2636
8.2434
.2234
.2035
.1837
.1640
8.1443
1'130 2630 3530 83°
tan
I
!
sin
,
O
60
449 59
446 58
443 57
440 56
.99437 55
434 54
431 53
428 52
424 51
.99421 60
418 49
415 48
412 47
409 46
.99406 45
402 44
399 43
396 42
393 41
.99390 40
386 39
383 38
380 37
377 36
.99374 36
370 34
367 33
364 32
360 31
.99357 30
354 29
351 28
I 347 27
344 26
.99341 26
337 24
334 23
331 22
327 21
.99324 20
320 19
317 18
314 17
310 16
.99307 15
303 14
300 13
297 12
293 11
.99290 10
9
286
283
8
279
7
276
6
.99272
5
269
4
265
3
262
2
1
258
.99255
0
7°
IV
I
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
36
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
56
56
57
58
59
60
sin
115
97° 187° 277°
cot
I
I
cos
.99255
.12187 .1227818.1443
251
308. 1248
216
248
338.1054
245
244
367 .0860
274
240
397 .0667
302
.12331 .12426 8.0476 .99237
233
456 .0285
360
230
485 8.0095
389
226
515 7.9906
418
222
544 .9718
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215
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211
633 .9158
533
208
662 .8973
562
204
692 .8789
591
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197
649
751 8424
193
678
781 .8243
189
810 .8062
706
186
840 .7882
735
.12764 .12869 7.7704 .99182
178
899 .7525
793
175
929 .7348
822
171
958 .7171
851
167
880 .12988 .6996
.12908 .13017 7.6821 .99163
160
047 .6647
937
156
076 .6473
966
152
106 .6301
.12995
148
136 .6129
. 13024
. 13053 .13165 7.5958 .99144
141
195 .5787
081
137
224 .5618
110
139 I 254 I .5449 I 133
129
168
28-1 .5281
.13197 .13313 7.5113 .99125
122
343 .4947
226
118
372 .4781
254
114
402 .4615
283
110
432 .4451
312
.13341 .13461 7.4287 .99106
102
491 .4124
370
098
521 .3962
399
094
427
550 .3800
091
580 .3639
456
.13485 .13609 7.3479 .99087
083
639 .3319
514
079
669 .3160
543
075
698 .3002
572
071
728 .2844
600
.13629 .13758 7.2687 .99067
063
787 .2531
658
059
817 .2375
687
846 .2220
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716
051
876 .2066
744
.13773 .13906 7.1912 .99047
043
935 .1759
802
039
965 .1607
831
035
860 .13995 .1455
031
889 .14024 .1304
.13917 .14054 I 7.1154 .99027
.
I
'
I
tan
COS
!
cot
!
i
82°
tan
I
sin
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30.
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
I
0
I
'
1720 2620 3520
~
980 1880 2780
sin
.13917
946
.13975
.14004
033
.14061
090
119
148
177
.14205
234
263
292
320
.14349
378
407
436
464
.14493
522
551
580
608
.14637
666
695
723
752
.14781
810
838
867
895,
TABLE
8°
tan
cot
.14054
084
113
143
173
. 14202
232
262
291
321
.14351
381
410
440
470
.14499
529
559
588
618
.14648
678
707
737
767
.14796
826
856
886
915
.14945
.14975
.15005
I 034
064
35
.14925
. 15094
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
954
.14982
.15011
040
.15069
097
126
155
184
.15212
241
270
299
327
.15351)
385
414
442
471
.15500
529
557
586
615
.15643
124
153
183
213
.15243
272
302
332
362
.15391
421
451
481
511
.15540
570
600
630
660
.15689
719
749
779
809
.15838
7.1154
.1004
.0855
.0706
.0558
7.0410
.0264
7.0117
6.9972
.9827
6.9682
.9538
.9395
.9252
.9110
6.8969
.8828
.8687
.8548
.8408
6.8269
.8131
.7994
.7856
.7720
6.7584
.7448
.7313
.7179
.7045
6.6912
.6779
.6646
.6514
6iR3
i6.6252
.6122
.5992
.5863
.5734
6.5606
.5478
.5350
.5223
.5397
6.4971
.4846
.4721
.4596
.4472
6.4348
.4225
.4103
.3980
.3859
6.3737
.3617
.3496
.3376
.3257
6.3138
cas!
cot
t:ln
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
1710
26103510
I
srru
cos
.99027
023
019
015
011
.99006
.99002
.98998
994
990
.98986
982
978
973
969
.98965
961
957
953
948
.98944
940
936
931
927
.98923
919
914
910
906
.98902
897
893
I 889
884
198880
876
871
867
863
.98858
854
849
845
841
.98836
832
827
823
818
.98814
809
805
800
796
.98791
787
782
778
773
.98769
sb.
I
go
IV
'
sin
tan
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
35
35
34
33
32
31
30
29
28
27
26
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
,j
.15643
672
701
730
758
.15787
816
845
873
902
.15931
959
.15988
.16017
046
.16074
103
132
160
189
.16218
246
275
304
333
.16361
390
419
447
476
.16505
533
562
591
.15838
868
891)
928
958
.15988
.16017
047
077
107
.16137
167
196
226
256
.16286
316
346
376
405
.16435
465
495
525
555
.16585
615
645
674
704
.16734
764
794
25
35
. 16648 I116884
24
23
22
21
2~
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
677
706
'14
,03
.16792
820
849
878
906
.16935
964
16992
.17021
050
.17078
107
136
164
193
.17222
250
279
308
336
.17365
A1n
cos
116
i
824
R",j
I
cot
I
cos
I
6.3138 .98769
.3019
764
.2901
760
.2783
755
.2666
751
6.2549 .98746
.2432
741
.2316
737
.2200
732
.2085
728
6.1970 .98723
.1856
718
.1742
714
.1628
709
.1515
704
6.1402 .98700
.1290
695
.1178
690
.1066
686
.0955
681
6.0844 .98676
.0734
671
.0624
667
.0514
662
.0405
657
6.0296 .98652
.0188
648
6.0080
643
5.9972
638
.9865
633
5.9758 .98629
.9651
624
.9545
619
94~~ I
I . (n,
,
!' 5.9228
914.9124
944.9019
.16974
8915
.17004
.8811
.17033 5.8708
063 .8605
093 .8502
123 .8400
153 .8298
.17183 5.8197
213 .8095
243 .7994
273 .7894
303 .7794
.17333 5 . 7694
363 .7594
393 .7495
423 .7396
453 .7297
.17483 5.7199
513 .7101
543 .7004
573 .6906
603 .6809
.17633 5.6713
i'
sin
{
57
581~1
56
:';."
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39 I -,
38
37
36
35
34
33
32
31
30
29
28
~A~
98604 25
600 24
595 23
590 22
585 21
.98580 20
575 19
570 .18
565 17
561 16
.98556 15
551 14
546 13
541 12
536 ]1
.98531 10
9
526
8
521
7
516
6
511
5
.98506
4
501
3
496
2
491
I
486
0
.98481
80°
1700 2600 3500
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
.33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
10°
tan
0 . 17365
60
59
n
cot
1000 1900 2800
990 1890 2790
5.6713
393
.6617
422
.6521
451
.6425
479
.6329
.17508
5.6234
537
. 6140
565
.6045
594
.5951
623
.5857
.17651
5.5764
680
.5671
708
.5578
737
.5485
766
.5393
.17794
5.5301
823
.5209
852
.5118
880
.5026
909
.4936
.17937
5.4845
966
.4755
. 17995
.4665
.18023
.4575
052
.4486
.18081
5.4397
109
.4308
138
. 4219
166
.4131
195
.4043
.18224
5.3955
252
.3868
281
.3781
I
309
624 .3694
I
I
338
64.
18367 I. 18684 i5.3521
395
714 .3435
424
745 .3349
452
775 .3263
481
805 .3178
.18509 .18835 5.3093
538
865 .3008
567
895 .2924
595
925 .2839
624
955 .2755
.18652 . 18986 5.2672
681 .19016 .2588
710
046 .2505
738
076 .2422
767
106 .2339
.18795 .191365.2257
824
166 .2174
852
197 .2092
881
227 .2011
910
257 .1929
.18938 .19287 5.1848
967
317 . 1767
.18995
347 . 1686
.19024
378 .1606
052
408 . 1526
.19081 .19438 5.1446
1
cos
I
cot
1690 2590 3490
cos
cot
. 17633
663
693
723
753
. 17783
813
843
873
903
.17933
963
. 17993
.18023
053
.18083
113
143
173
203
.18233
263
293
323
353
.18384
414
444
474
504
. 18534
564
594
I
79°
tan
110
TABLE IV
. 98481
476
471
466
461
.98455
450
445
440
435
.98430
425
420
414
409
.98404
399
394
389
383
.98378
373
368
362
357
.98352
347
341
336
I 331
.98325
320
315
I
sin
cot
cos
0 . 19U81 . 19438 5. 1446 .98163 60
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
I
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
310 27
198299
294
288
283
277
.98272
267
261
256
250
.98245
240
234
229
223
.98218
212
207
201
196
.98190
185
179
174
168
.98163
tan
1010 19102810
33
~
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
109
138
167
]95
.19224
252
281
309
338
. 19366
395
423
452
481
.19509
538
566
595.
623
. 19652
680
709
737
766
.19794
823
851
880
908
.19937
965
. 19994
468 .1366
157
498.
1286
152
529.1207
146
559.1128
140
.195895.1049
.98135
619 .0970
129
649 .0892
124
680.0814
118
710 .0736
112
. 19740 5 0658 .98107
770 .0581
101
801
.0504
096
831 .0427
090
861.0350
084
.19891 5.0273 .98079
921 .0197
073
952 .0121
067
19982 5.0045
061
.20012 4.9969
056
.20042 4.9894 .98050
073.9819
044
103. g744
039
133 .9669
033
164.9594
027
.20194 4.9520 .98021
224.9446
016
254 .9372
010
28'5.9298.98004
315.9225.97998
.20345 4.9152 .97992
376.9078
987
406.9006
981
.20022
436 .8933
975
OS!. I j(,)..
8860.I 969
.20079 i. 2~497 !4.8788 !. 97963
108
527 .8716
958
136
557.8644
952
165
588.8573
946
193
618.8501
940
.20222 .20648 4.8430 .97934
250
679 .8359
928
279
709.8288
922
307
739 .8218
916
336
770 .8147
910
.20364 .20800 4.8077 .97905
393
830.8007
899
421
861 .7937
893
450
891 .7867
887
478
921 .7798
881
1
I
50 .20507 .20952 4.7729 .97875
51
535.20982.7659
52
563 .21013
.7591
53
592
043 .7522
54
620
073.7453
65 .20649 .21104 4.7385
56
677
134 .7317
57
706
164.7249
58
734
195 .7181
59
763
225 .7114
60 .20791 .21256 4.7D46
sin
CDS
117
cot
tan
78°
869
863
857
851
.97845
839
833
827
821
.97815
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
25
24
23
22
21
20
19
18
17
16
15
14
13
12
II
10
9
8
7
6
5
4
3
2
1
0
sin
1680 2580 848~
1\
l
.W'
1020 1920 2820
,
O
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
sin
I
tan
12"
I
cos
cot
sin
.20791 1.21256 4.7046 .97815
809
.6979
286
820
803
.6912
848
316
797
347
.6845
877
791
.6779
377
905
.97784
.20933 .21408 4.6712
.6646
778
962
438
772
.6580
469
.20990
.6514
766
499
.21019
760
.6448
047
529
.97754
.21076 .21560 4.6382
.6317
748
104
590
.6252
742
621
132
.6187
735
161
651
.6122
729
682
189
.21218 .21712 4.6057 .97723
717
246
743
.5993
.5928
711
275
773
.5864
705
804
303
.5800
698
331
834
.21360 .21864 4.5736 .97692
686
.5673
388
895
.5609
680
417
925
673
445
956
.5546
.5483
667
474 .21986
.97661
.21502 .22017 4.5420
655
530
047
.5357
648
078
.5294
559
.5232
642
108
587
636
616
139 .5169
.21644 .22169 4.5107 .97630
623
.5045
672
200
.4983
617
701
231
611
.4922
729
261
I
604
292.4860
758 I
.217861.22322
14.4799 1.97598
.4737
592
353
814
585
.4676
843
383
.4615
579
414
871
573
.4555
444
899
.97566
.21928 .22475 4.4494
560
.4434
956
505
.4373
553
536
.21985
547
.4313
.22013
567
.4253
541
597
041
.97534
.22070 .22628 4.4194
.4134
658
528
098
.4075
521
126
689
719
.4015
515
155
.3956
508
183
750
.97502
.22212 .22781 4.3897
.3838
496
240
811
489
842
.3779
268
483
872
.3721
297
476
903
.3662
325
.97470
.22353 .22934 4.3604
964
.3546
463
382
.3488
457
410 .22995
450
.3430
438 .23026
444
056
.3372
467
.97437
.22495 .23087 4.3315
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
I
0
I
cos
I
cot
1670 257" 3470
I
13°
TABLE IV
77°
tan
sin
I
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
118
cot
tan
!
I
cot
I
76°
'J.
1040 1940 2840
,
cos
.22495 .23087 4.331.5 .97437
523
117
.3257
430
552
148.3200
424
580
179 .3143
417
608
209' .3086
411
.22637 .232404.3029.97404
665
271
.2972
398
693
301
.2916
391
722
332.2859
384
750
363.2803
378
.22778 .23393 4.2747
.97371
807
424.2691
36;
835
455.2635
358
863
485
.2580
351
892
516.2524
34;
.22920 .23547 4.2468 .97338
948
578
.2413
331
.22977
608.2358
32;
.23005
639.2303
318
033
670
.2248
311
.23062 .23700 4.2193 .97304
090
731
.2139
298
118
762.
2084
291
146
793.2030
284
175
823.1976
278
.23203 .23854 4. 1922 .97271
231
885.
1868
264
260
916
.1814
257
288
946.
1760
251
316.23977.1706
244
.23345 .24008 4.1653 .97237
373
039
.1600
230
401
069.1547
223
217
429
100 .1493
458
131 I .1441 ,
2m
.234861.241621413881.97203
514
193.
1335
196
542
223
.1282
189
571
254
.1230
182
599
285
.1178
176
.23627 .24-.164.1126.97169
656
347
.1074
162
684
377.1022
15;
712
408
.0970
]48
740
439
.0918
]41
.23769 .24470 4.0867
.97134
797
501
.0815
127
825
532.0764
120
853
562.0713
113
882
593
.0662
106
.23910 .246244.0611
.97100
938
655.0560
093
966
686.0509
086
.23995
717.0459
079
.24023
747.0408
072
.24051 .24778 4 0358 .97065
079
809.0308
058
108
840
.0257
051
136
871
.0207
044
164
902
.0158
037
.24192 .249334.0108.97030
cos
'
1030 1930288°
tan
I
sin
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
I
0
I'
1660 2560 3460
f
~i.
".t-:
""~~';~
.J6
.
.'
, '
, "¥:
'~...
"
,
,
'
.'t,{
- -
i.
-- ,-
.
.....
O
I
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
sin
.24192
220
249
277
305
.24333
362
390
418
446
.24474
503
531
559
587
.24615
644
672
700
728
.24756
784
813
841
869
.24897
925
954
.24982
.25010
.25038
066
094
122
151
.25179
207
235
263
291
.25320
348
376
404
432
.25460
488
516
545
573
.25601
629
657
685
713
.25741
769
798
826
854
.25882
cos
14°
tan
,
cos
.24933 4.0108
.97030
023
964
.0058
.24995 4.0009
015
.25026 3.9959
008
056
.9910 .97001
.25087 3.9861
.96994
987
118 .9812
149 .9763
980
180 .9714
973
211
966
.9665
.25242 3.9617
.96959
273
952
.9568
304
.9520
945
335
.9471
937
366
.9423
930
.25397 3.9375 .96923
428
.9327
916
459
.9279
909
490
.9232
902
52]
.9184
894
.25552 3.9136 .96887
583
.9089
880
.9042
614
873
645
.8995
866
.8947
676
858
.25707 3.8900
.96851
.8854
738
844
769
.8807
837
800
.8760
829
.8714
831
822
.25862 3.8667
.96815
.8621
807
893
924
.8575
800
95.5 I .8528 I
793
I1.25986
.8482
786
1
1
.260173.8436
.96778
.8391
048
771
.8345
764
079
.8299
756
110
.8254
749
141
.26172 3.8208
.96742
.8163
203
734
.8118
727
235
266
.8073
719
.8028
712
297
.26328 3.7983 .96705
359
.7938
697
.7893
690
390
.7848
421
682
452
.7804
675
.26483 3 . 7760 .96667
515
.7715
660
546
.7671
653
577
.7627
645
608
.7583
638
.26639 3.7539 .96630
670
.7495
623
.7451
701
615
733
.7408
608
764
.7364
600
.26795 ).7321
.96593
I cot
! tan I sin
165° 255° 345° 76°
15°
TABLE IV
cot
sin
o .25882
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
I
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
16
~16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
I
tl:J
910
938
966
.25994
.26022
050
079
107
135
.26163
191
219
247
275
.26303
331
359
387
415
.26443
471
500
528
556
.26584
612
640
668
696
.26724
752
780
808
836
.26864
892
920
948
.26976
.27004
032
060
088
116
.27144
172
200
228
256
.27284
312
340
368
396
.27424
452
480
508
536
.27564
cos
tan
1050 1950 2850
cot
cos
.26795 3.7321 .96593
826
.7277
585
857
.7234
578
888
.7191
570
920
.7148
562
.26951 3.7105 .96555
.7062
.26982
547
540
.27013
.7019
044
532
.6976
076
.6933
524
.27107 3.6891 .96517
138
.6848
509
502
169
.6806
201
.6764
494
232
.6722
486
.27263 3.6680 .96479
471
294
.6638
326
.6596
463
357
.6554
456
.6512
448
388
.96440
.27419 3.6470
451
.6429
433
482
.6387
425
.6346
417
513
545
.6305
410
.27576 3.6264
.96402
607
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394
638
.6181
386
670
.6140
379
701
.6100
371
.27732 3.6059 .96363
764
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355
795
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347
.5937 I 340
I 826
858.5897
332
1.27889 13.5856 1.96324
921
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077
277
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269
140
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261
172
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234
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266
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230
297
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214
329
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391
198
423
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549
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158
580
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150
612
.4951
142
643
.4912
134
.28675 3.4874 .96126
I
cot
tan
74°
sin
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46'
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
I
0
!'
1640 254° 344°
~
1060 ]960 2860
16°
sin
I
0
1
2
3
4
5
6
7
8
9
10
11
12
13
]4
15
]6
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
H
34
36
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
5]
52
53
54
55
56
57
58
59
60
I
tan
TABLE
cot
I
cos
.27564 .286753.4874.96126
592
706.4836
118
620
738 .4798
1]0
] 02
648
769 .4760
676
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.4722
094
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731
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787
927.4570
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815
958 .4533
054
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871.29021.4458
037
899
053.4420
029
927
084 .4383
021
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039
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400 .4014
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432.3977
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290
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318
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346
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CQS
cot
I tan!
163° 253° 343°
73°
j
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2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
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20
21
22
23
24
25
26
27
28
29
30
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
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30
29
28
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26
25
24
23
22
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20
19
18
17
16
15
14
13
12
1
10
9
8
7
6
5
4
3
2
]
17°
IV
1070 1970 287.
sin I tan I cot I cos I
.29237 .30573 3.2709 .95630 60
265
605.2675
622 59
293
637.2641
613 58
321
669 .2607
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404
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I
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sin
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cot
tan!
72°
363
354
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275
260
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1
29
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11
10
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8
7
6
5
4
3
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sin
1620 252° 34200
1080 ]980 2880 18°
,
I
sin
I
tan
I
TABLE IV
cot
0 .30902 .32492 3.0777
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cos
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.95106 60
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3 .30985
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36
36
37
38
39
40
41
42
43
44
46
46
47
48
49
60
51
52
53
54
55
56
57
58
59
60
I
I.33621
56
55
54
53
52
51
60
49
48
47
46
46
44
43
42
41
40
39
38
37
36
35
34
33
32
3]
30
29
28
27
26
[2.9743 .94786 25
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5
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2
400.9070
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1
.34433 2.9042 .94552
0
cot
1610 25]03410
1:1:1
71°
19°
sin
tan
1090 ]990 2890
cot
cos
I
0.32557.34433
2.9042 .94552 60
1
584
465.9015
542 59
2
612
498 .8987
533 58
3
639
530 .8960
523 57
4
667
563 .8933
514
5.32694.34596
2.8905 .94504
6
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495
7
749
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8
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693.8824
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428
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16.32997
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380
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20 .33106 .35085 2.8502 .9436]
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183.8423
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216.8397
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25.33244.35248
2.8370 .94313
26
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346.8291
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1
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
36
34
33
32
31
30
29
28
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26
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6
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5
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3
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2
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364.7500
979
I
59
60 .34202 .36397 2.7475 .93969
0
cos
sin
121
!
cot
10°
tan
1600
sh
,
2500 3400
2000 2900
1100
sin
o
I
2
3
4
6
6
7
8
9
10
11
12
13
14
16
16
17
18
19
20
21
22
23
24
26
26
27
28
29
30
31
32
33
34
36
36
37
38
39
40
41
42
43
44
46
46
47
48
49
60
51
52
53
54
66
56
57
58
59
60
20°
tan
cot
.34202
229
257
284
311
.34339
366
393
421
448
.34475
503
530
557
584
.34612
639
666
694
721
.34748
775
803
830
857
.34884
912
939
966
.34993
.35021
.36397 2.7475
430 .7450
463 .7425
496 .7400
529 .7376
.36562 2.7351
595 .7326
628 .7302
661 .7277
694 .7253
.36727 2.7228
760 .7204
793 .7179
826 .7155
859 .7130
.36892 2.7106
925 .7082
958 .7058
.7034
.36991
.37024
.7009
.37057 2.6985
090 .6961
123 .6937
157 .6913
190 .6889
.37223 2.6865
256 .6841
289 .6818
322 .6794
355 .6770
.37388 2.6746
422 .6723
048
075,
455,.6699
102 1 488 1 .6675
521 .6652
130 i
.35157 .37554 2.6628
588 .6605
184
621 .6581
211
654 .6558
239
687 .6534
266
.35293 .37720 2.6511
754 .6488
320
347
787 .6464
820 .6441
375
853 .6418
402
.35429 .37887 2.6395
920 .6371
456
953 .6348
484
.6325
511 .37986
.6302
538 .38020
.35565 .38053 2.6279
086 .6256
592
120 .6233
619
153 .6210
647
186 .6187
674
.35701 .38220 2.6165
253 .6142
728
286 .6119
755
320 .6096
782
353 .6074
810
.35837 .38386 2.6051
I
I
CDS
cot
I
1690 2490 339° 69°
1120 2020 2920
TABLE IV
tan
cos
.93969
959
949
939
929
.93919
909
899
889
879
.93869
859
849
839
829
.93819
809
799
789
779
.93769
759
748
738
728
.93718
708
698
688
677
.93667
657
647
1 637
1 626
.93616
606
596
585
575
.93565
555
544
534
524
.93514
503
493
483
472
.93462
452
441
431
420
.93410
400
389
379
368
.93358
sin'
sin
0
1
2
3
4
6
6
7
8
9
10
11
12
13
14
16
16
17
18
19
20
21
22
23
24
26
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
46
46
47
48
49
60
51
52
53
54
65
56
57
58
59
60
60
59
58
57
56
65
54
. 53
52
51
60
49
48
47
46
46
44
43
42
41
40
39
38
37
36
36
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
16
14
13
12
11
10
9
8
7
6
5
4
3
2
I
0
1
122
tan
cot
G8°
cot
'Isinltan
cot
.35837 .38386 2.6051 .93358
420 .6028
348
864
453 .6006
337
891
487 .5983
327
918
945
520 .5961
316
. 35973 .38553 '2.5938 .93306
.36000
587 .5916
295
620 .5893
285
027
654 .5871
274
054
687 .5848
264
081
.36108 .38721 2.5826 .93253
754 .5804
243
135
162
787 .5782
232
821 .5759
222
190
217
854 .5737
211
. 36244 .38888 2.5715 .93201
92 I . 5693
190
271
955 .5671
180
298
325 . 38988 .5649
169
352 .39022 .5627
159
.36379 .39055 2.5605 .93148
089 .5583
137
406
434
122
.5561
127
461
156 .5539
116
190 .5517
106
488
.36515 .39223 2.5495 .93095
542
257 .5473
084
569
290 .5452
074
324 .5430
063
596
623
357 .5408
052
.36650 .39391 2.5386 .93042
425 .5365
031
677
458 .5343
020
704
731 i 492 i .5322 i. 93010
526 i .5300 .92999
758
.36785 .39559 2.5279 .92988
593 .5257
978
812
626 .5236
967
839
\ 660 .5214
956
867
694 .5193
945
894
. 36921 .39727 2.5172 .9293;
761 .5150
924
948
795 .5129
913
.36975
829 .5108
902
.37002
862 .5086
892
029
.37056 .39896 2.5065 .92881
930 .5044
870
083
963 .5023
859
110
849
137 .39997 .5002
838
164 .40031 .4981
.37191 .4006; 2.4960 .92827
098 .4939
816
218
132 .4918
805
245
166 .4897
794
272
200 .4876
784
299
.37326 .40234 2.485; .92773
267 .4834
762
353
301 .4813
751
380
33; .4792
740
407
369 .4772
729
434
.37461 .40403 2.4751 .92718
cos
.'
tan
220
60.
59 ."
58
:'
57
56 -.;'~'
65X~{1
54 ':; '
..;..
,
,
53 ':, '
52 .",,"
51 ~:;.
:
60 .t..,;
49 'J
48.;~~~
47 :'.;
46:~
45
44
'
"
,,
,-
,_,.-.
43
42"
41
40
39
38
37
1
36
35
34
33
32
31
30 ':
29
28-1
.
'/
.,~
!~
.
8
'~-"
'
.
;: I .,~~
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
sin
1680 2480 338°
.1
:
cos
TABLE IV
,
O .37461 .40403 2.4751 .92718
707
436 .4730
488
1
697
51;
470 .4709
2
686
504 .4689
542
3
675
538 .4668
569
4
5 .37595 .40572 2.4648 .92664
653
606 .4627
622
6
642
640 .4606
649
7
631
676
674 .4586
8
620
707 .4566
703
9
10 .37730 .40741 2.4545 .92609
598
775 .45.::5
757
11
587
809 .4504
784
12
576
843 .4484
811
13
565
877 .4464
838
14
16 .37865 .40911 2.4443 .92554
543
945 .4423
892
16
532
.4403
919 .40979
17
521
.4383
946 .41013
18
510
047 .4362
973
19
20 .37999 .41081 2.4342 .92499
488
115 .4322
21 .38026
477
149 .4302
053
22
466
183 .4282
080
23
455
217 .4 262
107
24
26 .38134 .41251 2.4242 .92444
432
285 .4222
161
26
421
319 .4202
188
27
410
353 .4182
215
28
399
387 .4162
241
29
30 .38268 .41421 2.4142 .92388
377
455 .4122
295
31
366
490 .4102
322
32
349 i 524 i .4083 i 355
33
343
34
376'
558 i .4063,
35 .38403 .41592 '2.4043 1.92332
321
626 .4023
430
36
310
660 .4004
456
37
299
694 .3984
483
38
287
728 .3964
510
39
40 .38537 .41763 2.394; .92276
265
797 .3925
564
41
254
.3906
831
591
42
243
865 .3886
617
43
23\
899 .3867
644
44
46 .38671 .41933 2.3847 .92220
209
.3828
698 .41968
46
198
.3808
72; .42002
47
186
036 .3789
752
48
175
070 .3770
778
49
60 .38805 .4210; 2.3750 .92164
152
832
51
1391.3731
141
173 .3712
859
52
130
207 I .3693
886
53
119
.3673
242
912
54
.92107
65 .38939 .42276,2.3654
096
310 .363;
966
56
085
345 .3616
57 .38993
073
379 .3597
58 .39020
062
413 .3578
046
59
60 .39073 .42447 2.3559 .92050
I
-
J ,cos I
cot
I
1670 2470 3370 67°
tan
sin
60
59
58
57
56
66
54
53
52
51
60
49
48
47
46
45
44
43
42
41
40
39
38
37
36
36
34
33
32
3\
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
6
4
3
2
1
0
0
1
2
3
4
6
6
7
8
9
10
II
12
13
14
16
16
17
18
19
20
21
22
23
24
26
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
66
56
57
58
59
60
230
123
2030 2930
60
59
58
57
56
66
54
53
52
51
60
49
48
47
46
46
44
43
42
41
40
39
38
37
36
36
34
33
32
31
30
29
28
27
26
.40008 i. 43654 12.2907 1.91648 26
636 24
689 .2889
035
625 23
724 .2871
062
613 22
758 .2853
088
601 21
793 .2835
115
.40141 .43828 2.2817 .91590 20
578 19
862 .2799
168
566 18
897 .2781
195
555 17
932 .2763
221
543 16
248 .43966 .2745
.4027; .44001 2.2727 .91531 16
519 14
036 .2709
301
508 13
071 .2691
328
496 12
35;
105 .2673
484 11
140 .2655
381
.40408 .4417; 2.2637 .91472 10
461 9
210 .2620
434
449 8
244 .2602
461
437 7
279 .2584
488
425 6
314 .2566
514
.40541 .44349 2.2549 .91414 5
402 4
384 .2531
567
390 3
418 .2513
594
378 2
453 .2496
621
366 1
488 .2478
647
0
.40674 .44523 2.2460 .9135;
cos
'
1130
COB
cot
tan
sin
.39073 .42447 2.3559 .92050
039
482 .3539
100
028
516 .3520
127
016
551 .3501
153
585 .3483 .92005
180
.39207 .42619 2.3464 .91994
982
654 .3445
234
971
688 .3426
260
959
722 .3407
287
948
757 .3388
314
.39341 .42791 2.3369 .91936
925
826 .3351
367
914
860 .3332
394
902
894 .3313
421
891
929 .3294
448
.39474 .42963 2.3276 .91879
868
501 .42998 .3257
856
528 .43032 .3238
845
067 .3220
555
833
101 .3201
581
.39608 .43136 2.3183 .91822
810
170 .3164
635
799
205 .3146
661
787
239 .3127
688
775
274 .3109
715
.39741 .43308 2.3090 .91764
752
343 .3072
768
741
378 .3053
795
729
412 .3035
822
718
447 .3017
848
.39875 .43481 2.2998 .91706
694
516 .2980
902
683
928
550 .2962
955,
585, .2944
671
.39982 1 620 I 2925 I 660
tan
cot
66°
I
sin
1560 2460 3360
24°
TABLE IV
25° 1150 2050 2950
sin
tan
cot
cos
sin I tan I cot I cos
I
0 .40674 .44523 2.2460 .91355 60
0 .42262 .46631 2. 1445 .90631 60
I
700
558.2443
343 59
I
288
666. 1429
618 59
2
727
593.2425
331 58
2
315
702 .1413
606 58
3
753
627 .2408
319 57
3
341
737 .1396
594 57
4
780
662.2390
307 56
4
367
772. 1380
582 56
5 .40806 .44697 2.2373 .91295 55
5 .42394 .46808 2. 1364 .90569 55
6
833
732.2355
283 54
6
420
843.13218
557 54
7
860
767 .2338
272 53
7
446
879. 1332
545 53
8
886
802.2320
260 52
8
473
914 .1315
532 52
9
913
837 .2303
248 51
9
499
950 .1299
520 51
10 .40939 .44872 2.2286 .91236 50
10 .42525 .46985 2.1283 .90507 50
. 907 .2268
11
966
224 49
II
552.47021.
1267
495 49
12 .40992
942.2251
212 48
12
578
056 .1251
483 48
13 .41019 .44977 .2234
200 47
13
604
092.1235
470 47
14
045 .45012 .2216
188 46
14
631
128. 1219
458 46
15 .41072 .45047 2.2199 .91176 45
15 .42657 .47163 2.1203 .90446 45
16
098
082.2182
164 44
16
683
199 .1187
433 44
17
125
117.2165
152 43
17
709
234 .1171
421 43
18
151
152 .2148
140 41
18
736
270 .1155
408 42
19
178
187.2130
128 41
19
762
305.1139
396 41
20 .41204 .452222.2113 .91116 40
20 .42788 .47341 2. 1123 .90383 40
21
231
257 .2096
104 39
21
815
377 .1107
371 39
22
257
292.2079
092 38
22
841
412. 1092
358 38
23
284
327..2062
080 37
23
867
448.1076
346 37
24
310
362.2045
068 36
24
894
483. 1060
334 36
25 .41337 .453972.2028 .91056 35
25 .42920 .47519 2.1044 .90321 35
26
363
432.2011
044 34
26
946
55;.
1028
309 34
27
390
467. 1994
032 33
27
972
590. 1013
296 33
28
416
502. 1977
020 32
28 .42999
626 0997
284 32
29
443
538 .1960 .91008 31
29 .4302;
662 .0981
271 31
30 .41469 .45573 2.1943 .90996 30
30 .43051 .47698 2.0965 .90259 30
31
496
608. 1926
984 29
31
077
733.09;0
246 29
32
522
643. 1909
972 28
32
104
769.0934
233 28
33
549
678 .1892
960 27
33
130
805 .0918
221 27
34
575 I 71~. I .1876. I 948 26
34
156 i 840 i .0903
208 26
35 .4160245748 '2.1859 1.90936 25
35 .431821.47876 '2.0887 .90196 25
36
628
784. 1842
924 24
36
209
912.0872
183 24
37
655
819 .1825
911 23
37
23;
948 .0856
171 23
38
681
854. 1808
899 22
38
261.47984.0840
J58 22
39
707
889. 1792
887 21
39
287 .48019 \ .0825
146 21
40 .41734 .45924 2.1775 .90875 20
40 .43313 .48055 2.0809 .90133 20
41
760
960. 1758
863 19
41
340
091 .0794
120 19
42
787 .45995 .1742
851 18
42
366
127 .0778
108 18
43
813 .46030 . 1725
839 17
43
392
163 .0763
095 17
44
840
065.1708
826 16
44
418
198.0748
082 J6
45 .4I866 .46101 2. 1692 .90814 15
45 .43445 .48234 2.0732 .90070 15
46
892
136. 1675
802 14
46
471
270 .0717
057 14
47
919
171 .1659
790 13
47
497
306 .0701
045 13
48
945
206. 1642
778 12
48
523
342.0686
032 12
49
972
242 .1625
766 11
49
549
378 .0671
019 II
50 .41998 .462772.1609 .90753 10
50 .43575 .48414 2.0655 .90007 10
51 .42024
312 .1592
741 9
51
602
4;0.0640.89994
9
52
051
348 .1576
729 8
52
628
486.0625
981 8
53
077
383 .1560
717 7
53
654
521 .0609
968 7
54
104
418. 1543
704 6
54
680
557.0594
956 6
55 .42130 .46454 2.1527 .90692 5
55 .43706 .48593 2.0579 .89943 5
56
156
489. 1510
680 4
56
733
629.0564
930 4
57
183
525 .1494
668 3
57
759
665.0549
918 3
58
209
560. 1478
655 2
58
78;
70I .0533
905 2
59
235
595 .1461
643
1
59
811
737 .0518
892
I
60 .42262 .46631 2.1445 .9063I
0
60 .43837 .48773 2.0503 .89879 0 I
1140 2040 2940
1
1
cos
,
cot
1550 2450 3350
I
tan'
1160 2060 2960
sin
I
:!-24
6~O
1Q4,°244Q 334°
26°
I
.
~
I
I
cos
'I
.89879 60
867 59
854 58
841 57
828 56
.89816 55
803 54
790 53
777 52
764 51
.89752 50
739 49
726 48
713 47
700 46
.89687 45
674 44
662 43
649 42
63641
.89623 40
610 39
597 38
584 37
571 36
.89558 35
545 34
532 33
519 32
506 31
.89493 30
480 29
467 28
454 27
776
076.9970
415
37
802
113.9955
402
38
828
149 .9941
389
39
854
185.9926
376
40 .44880 .50222 1.9912 .89363
41
906
258.9897
350
42
932
295.9883
337
43
958
331 .9868
324
44 .44984
368.9854
311
45.45010.504041.9840.8929815
46
036
441 .9825
285
47
062
477 .9811
272
48
088
514.9797
259
49
114
550.9782
245
50 .45140 .50587 1.9768 .89232
51
166
623.9754
219
52
192
660.9740
206
53
218
696 9725
193
54
243
733.9711
180
55 .45269 .50769 1.9697 .89167
56
295
806.9683
153
57
321
843.9669
140
58
347
879.9654
127
59
373
916.9640
114
60 .45399 .50953 1. 9626 .89101
153e 243°~33
sin
1170 2070 2970
cot
I
cos
14
278 I 205 9155
647
35.46304 i. 52242 119142 1.88634
36
330
279 I .9128
620
37
355
316.9115
607
38
381
353 .9101
593
39
407
390.9088
580
40.46433.52427
1.9074 .88566
41
458
464.9061
553
42
484
501 9047
539
43
510
538 9034
526
44
536
575.9020
512
45.46561.526131.9007.8849915
46
587
650.8993
485
47
613
687 .8980
472
48
639
724.8967
458
49
664
761 .8953
445
50.46690.52798
1.8940 .88431
51
716
836.8927
417
52
742
873.8913
404
53
767
910 .8900
390
54
793
947.8887
377
55.46819.52985
1.8873 .88363
56
844.53022.8860
349
57
870
059.8847
336
58
896
096.8834
322
59
921
134.8820
308
60 .46947 .53171 1.8807 .88295
24
23
22
21
20
19
18
17
16
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
tan!sinl'
Icosl
----------------
I
tan
I
0 .45399 .50953 1.9626 .89101 60
1
42;.50989.9612
087 59
2
451.51026.9598
074 58
3
477
063 .9584
061 57
4
503
.099
.9570
048 56
5 .45529 .51136 1.9556 .8903;
55
6
554
173 .9542
021 54
7
580
209.9528.89008
53
8
606
246.9514.88995
52
9
632
283.9;00
981 51
10.45658.51319
1.9486 .88968 50
11
684
356.9472
955 49
12
710
393.9458
942 48
13
736
430 .9444
928 47
14
762
467 .9430
915 46
15.45787.51503
1.9416 .88902 45
16
813
540.9402
888 44
17
839
577.9388
875 43
18
865
614 .9375
862 42
19
891
651 .9361
84841
20.45917.51688
1. 9347 .88835 40
21
942
724 .9333
822 39
22
968
761 .9319
808 38
23.45994
798.9306
795 37
24.46020
835.9292
782 36
25.46046.51872
1.9278 .88768 35
26
072
909 .9265
755 34
27
097
946.9251
741 33
28
123.51983.9237
728 32
29
149.52020.9223
715 31
30 .46175 .52057 1.9210 .88701 30
31
201
094.9196
688 29
32
226
131.9183
674 28
16§
.9169
661 27
33
252
34
724 50004 I 9999 I 441 26
35 .44750 i. 50040 11.9984 i. 89428 25
36
27°
TABLE IV
cot
0 .43837 .48773 2.0503
1
863
809.0488
2
889
845.0473
3
916
881 .0458
4
942
917.
C443
5 .43968 .48953 2.0428
6.43994.48989
.0413
7 .44020 .49026
.0398
8
046
062.0383
9
072
098 .0368
10 .44098 .49134 2.0353
11
124
170.0338
12
151
206 .0323
13
177
242 .0308
14
203
278 .0293
15 .44229 .49315 2.0278
16
255
351 .0263
17
281
387.0248
18
307
423 .0233
19
333
459 .0219
20 .44359 .49495 2.0204
21
385
532 .0189
22
411
568.0174
23
437
604.0160
24
464
640 .0145
25 .44490 .49677 2.0130
26
516
713 .0115
27
542
749.0101
28
568
786.0086
29
594
822.0072
30 .44620 .49858 2.0057
646
894 .0042
31
32
672
931 .0028
33
698 .49967 2.0013
cos!cotl
sin
tan
125
cotltan!
26
25
24
23
22
21
20
19
18
17
16
14
13
12
11
10
9
8
7
6
5
4
3
2
I
0
~
sin
TABLE IV
tan
cot
J
29°
sin
I
119020902990
tan
cot
I
0 .46947 .53171 1.8807 .88295 60
1
973
208.8794
281 59
2 .46999
246.8781
267 58
3 .47024
283. 876~
254 57
4
050
320 .8755
24056
5.47076.533581.8741.88226
55
6
101
395 .8728
21354
7
127
432 .8715
199 53
8
153
470.8702
185 52
9
178
507 .8689
172 51
10 .47204 .53545 1.8676 .88158 50
]I
229
582.8663
144 49
12
255
620.86;0
]3048
13
281
657 .8637
117 47
0 .48481 .55431 1.8040 .87462 60
1
506
469.8028
448 59
2
532
507.8016
434 58
3
557
545.8003
420 57
4
583
583 .7991
40656
5.48608.556211.7979.87391
55
6
634
659 .7966
37754
7
659
697 .7954
363 53
8
684
736.7942
349 52
9
710
774 .7930
3355]
10 .48735 .55812 1.79(7 .87321 50
11
761
850.7905
306 49
12
786
888.7893
292 48
13
811
926 .7881
278 47
15.47332.537321.8611.8808945
16
358
769.8598
17
383
807 .8585
18
409
844 .8572
19
434
882 .8559
15.48862.560031.7856.8725045
16
888
041 .7844
235
17
913
079 .7832
221
18
938
117 .7820
207
19
964
156 .7808
193
20 .48989 .56194 1.7796 .87178
21.49014
232 .7783
164
22
040
270.7771
150
23
065
309.7759
136
24
090
347.7747
121
14
306
694.8624
103 46
44
43
42
4]
.47460 .53920 1.8546 .88020 40
486
95Z .8533.88006
39
511.53995.8520.87993
38
537.54032.8507
972 37
562
070.8495
965 36
20
21
22
23
24
075
062
048
034
25 .47588 .54107 I. 8482 .87951 35
26
614
145 .8469
937 34
27
632
183.8456
92333
28
665
220.8443
909 32
29
690
258.8430
896 31
30 .47716 .54296 1.8418 .87882 30
31
741
333 .8405
868 29
32
767
371 .8392
854 28
i .'
i
522.8341
i
I
34
818
36
862
i
446 .8367:
826 26
35 .47844 .54484 :1.8354 .87812 25
1
37
895
560.8329
38
920
597.8316
39
946
635.8303
40.47971.546731.8291.8774320
41 .47997
711 .8278
42 .48022
748 .8265
43
048
786 .8253
44
073
824.8240
798
24
14
264 46
949
518
504
4
3
56
57
58
430
355.8065
490
2
59
456
393.8053
476
GO .48481 .55431 I. 8040 .87462
1
0
I
cot
1610 2410 3310
tan
61 °
I
sin
I
'
:
899
924
7367
661
7355
646
58
950
657' 7344
632
59.49975
696:
7332
617
60.50000.57735
1. 7321 .86603
cos
1~6
352
442 49
42748
413 47
279.7159
1\
12
13
398 46
580
619'
cot
tan
60°
I
sin
1600 2400
4
3
2
1
0
I
'
3300..&
cos
21
22
23
24
528
553
578
603
369
354
340
325
552 .7079
591 .7067
631 .7056
670.7045
37
929
38
954
39 .50979
40 .51004
41
029
42
054
43
079
44
104
26
25
I
46
154
533 .6797
926
47
179
573 .6786
911
48
204
612 .6775
896
49
229
651 .6764
881
50 .5.1254 .59691 1.6753 .85866
51
279
730 .6742
851
52
304
770.6731
836
53
329
809 .6720
821
54
354
849.6709
806
55 .51379 .59888 1.6698 .85792
14
13
12
11
10
9
8
7
6
5
56
404
928.6687
777
57
429.59967.6676
762
58
454.60007.6665
747
046 .6654
732
59
479
60 .51504 .60086 11. 6643 .85717
4
3
2
1
0
!
cot
'I
1490 2390 3290
I
tan
590
!
sin
I
551 49
53648
521
506 47
46
15.51877.606811.6479.8549\
16
902
72\ .6469
17
927
761 .6458
18
952
801 .6447
19 .51977
841 .6436
45
476 44
461 43
446 42
431 41
852
642.6490
21
22
23
24
24
179.6898
059 23
218.6887
045 22
258.6875
030 21
.592971.6864.8601520
336 .6853 .86000 19
376 .6842 .85985 18
415 .6831
970 17
454.6820
956 16
\
778
803
828
20 .52002 .60881 1.6422 .85416 40
295 39
281 38
266 37
251 36
45 .51129 .59494 1.6808 .85941 15
cos
522.6523
562.6512
602 .6501 .
14
45
44
43
42
41
34
854:
061 .6932'
104
35 .50879 1.59101 11.6920 !. 86089
.6909
074
36
904
14)
279.8090
317.8078
I
162.7193
201 .7182
240 .7170
I
379
405
cos
14
277
302
327
34
344,
731 7627 i 978
35.49369.56769
I 7615 . 86964
7603
\
10 .51753 .60483 1.6534 .85567 50
1]
12
13
25 .50628 .58709 1.7033 .86237 35
808:
1210211" 3010
cot
10 .50252 .58124 1.7205 .86457 50
26
654
748 .7022
222 34
27
679
787.7011
20733
28
704
826.6999
192 32
29
729
865.6988
178 31
30 .50754 .58905 1.6977 .86163 30
31
779
944 .6965
148 29
32
804.58983.6954
133 28
394
tan
0 .51504 .60086 1.6643 .85717 60
1
529
126.6632
702 59
2
554
165.662 I
687 58
3
579
205.6610
672 57
4
604
245 .6599
65756
5.51628.602841.6588.85642
55
6
653
324 .6577
62754
7
678
364 .6566
612 53
8
703
403. 655~
597 52
9
728
443 .6545
58251
1.7735 87107 35
26=.
25
24
31°
sin
0 .50000 .57735 1.7321 .86603 60
I
025
774 .7309
588 59
2
050
813.7297
573 58
3
076
851 .7286
559 57
4
101
890 .7274
54456
5.50126.579291.7262.86530
55
6
151.57968
.7251
51554
7
176.58007
.7239
501 53
8
201
046.7228
486 52
9
227
085 .7216
471 51
20 .50503 .58513 1.7090 .86310 40
39
38
37
36
I
I
cos
40
55
57
45 .48099 .54862 1.8228 .87673 15
TABLE IV
cot
44
43
42
41
46
1~4
900 .8215
659 14
47
150
938 .8202
645 13
48
175.54975
.8190
631 12
49
201.55013
.8177
61711
50 .48226 .55051 1.816; .87603 10
51
252
0139 .8152
589 9
52
277
127.8140
575 8
53
303
165 .8127
561 7
54
328
203.8115
546 6
55 .48354 .55241 1.8103 .87532 5
19
18
17
16
30°
I
tan
15.50377.583181.7147.86384
16
403
357.7136
17
428
396 .7124
18
453
435 .7113
19
478
474 .7102
37
419
846 7591
935 23
38
445
885: 7579
921 22
39
470
9; \ .7567
906 21
40 .49495.569621.755686892
20
41
521 57000 7544'
878 19
42
546'
039 '7532
863 18
43
571
078 '7520
849 17
] 16 7508
44
596
834 16
45 .49622 5715; 1. 7496 . 86820 15
46
647'
193 7485
805 14
47
672
232 )473
791 13
48
697
271 7461
777 12
49
723
309 :7449
76211
50 .49748 .57348 1.7437 86748 10
51
773
386 7426'
733 9
52
798
425 :7414
719 8
53
824
464 7402
704 7
54
849
503: 7391
690 6
55 .49874 .57541 1.7379 .86675 5
729
715
701
687
sin
26
141
424 .7723'
093 34
27
166
462.7711
07933
28
192
501 .7699
064 32
29
217
539.7687
050 31
30 .49242 .56577 1.7675 87036 30
31
268
616 .7663'
021 29
32
293
654.7651.87007
28
36
784 23
770 22
756 21
837.55964.7868
25.49116.56385
120021003000
cos
026
921 .6415
401 39
051.60960.6404
385 38
076.61000.6393
370 37
101
040.6383
355 36
25.52126.61080 1. 6372 .8534Q 35
26
151
120 .6361
325 34
27
175
160.6351
31033
28
200
200.6340
294 32
29
225
240.6329
279 31
30 5225Q .61280 1.6319 .85264 30
31
275
320 .6308
249 29
32
299
360.6297
234 28
33
324,
400 i .6287 i
218 27
34
349
35.52374
440 I .
61480 1.626~ 185188
36
399
520.6255
173
37
423
561 .6244
157
38
448
601 .6234
142
39
473
641 .6223
127
40 .52408 .616811.6212.85112
41
522
721 .6202
096
42
547
761 .619\
081
43
572
801 .6181
066
44
597
842.6170
051
25
24
23
22
21
20
19
18
17
16
45 .52621 .61882 1.6160 .85035 15
46
646
922 .6149
02Q
47
671 .61962 .6139 .85005
48
696.62003
.6128.84989
49
720
043 .6118
974
50 .52745 .62083 .6107 .84959
51
770
124 .6097
943
52
794
164.6087
928
53
819
201 .6076
913
54
844
245.6066
897
.84882
55 .52869 .62285 1.605~ '
56
893
57
918
58
943
59
967
60.52992.62487
'
127
cos
325.6045
866
366.6034
851
406.6024
836
446.
6014
82~
1. 6003 .84805
1
118020802988 28°
cot
58°
tan
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
sin
1480 2380 328Q
~
32°
1220 2120 3020
,
I
I
sin
tan
O.52992
.62487
1 .53017
527
2
041
568
3
066
608
4
091
649
5 .53115 .62689
6
140
730
7
164
770
8
189
811
9
214
852
10 .53238 .62892
11
263
933
12
288 .62973
13
312 .63014
14
055
337'
15 .53361 .63095
386
16
136
411
17
177
18
435
217
460
19
258
20 .53484 .63299
21
509
340
534
22
380
23
558
421
24
583
462
25 .53607 .63503
632
26
544
27
656
584
681
28
625
705
29
666
30 .53730 .63707
754
31
748
779
32
789
804
33
830
828
871
34
35 .53853 163912
877
953
36
902 .63994
37
926 .64035
38
951
076
39
40 .53975 .64117
41 .54000
158
024
199
42
049
240
43
073
281
44
45 .54097 .64322
122
363
46
146
404
47
171
446
48
195
487
49
50 .54220 .64528
244
569
51
269
610
52
293
652
53
317
693
54
55 .54342 .64734
366
775
56
817
391
57
415
858
58
440
899
59
60 .54464 .64941
I
I
1470
cos
I
cot
2370 3270
1.6003
.5993
.5983
.5972
.5962
1.5952
.5941
.5931
.5921
.5911
1.5900
.5890
.5880
.5869
.5859
1.5849
.5839
.5829
.5818
.5808
1.5798
.5788
.5778
.5768
.5757
1.5747
.5737
.5727
.5717
.5707
1.5697
.5687
.5677
.5667
.5657
!15647
.5637
.5627
.5617
.5607
1.5597
.5587
.5577
.5567
.5557
1.5547
.5537
.5527
.5517
.5507
1.5497
.5487
.5477
.5468
.5458
1.5448
.5438
.5428
.5418
.5408
1.5399
I
tan
57°
330
TABLE IV
cot
0
I
2
3
4
5
6
7
8
9
10
II
sin
.54464
488
513
537
561
.54586
610
635
659
683
.54708
732
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
756
781
805
.54829
854
878
902
927
.54951
975
.54999
.55024
048
.55072
097
121
145
169
.55194
218
242
266,
291!
.55315!
339
363
388
412
.55436
460
484
509
533
.55557
581
605
630
654
.55678
702
cos
.84805
789
774
759
743
.84728
712
697
681
666
.84650
635
619
604
588
.84573
557
542
526
511
.84495
480
464
448
433
.84417
402
386
370
355
.84339
324
308
292
i
I
277
184261
245
230
214
198
.84182
167
151
135
120
.84104
088
072
057
041
.84025
.84009
.83994
978
962
.83946
930
915
899
883
.83867
I
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
I
0
52
'
128
I
.64941
.64982
.65024
065
106
.65148
189
231
272
314
.65355
397
I
cot
438 .5282
480 .5272
521 .5262
.65563 1.5253
604 .5243
646 .5233
688 .5224
729 .5214
.65771 1.5204
813 .5195
854 .5185
896 .5175
938 .5166
.65980 1.5156
.66021
.5147
063 .5137
105 .5127
147 .5118
.66189 1.5108
230 .5099
272 .5089
314 . 5080
35h
5070
66398 r1.5061
440 .5051
482 .5042
524 .5032
~66 .5023
.66608 1.5013
650 .5004
692 .4994
734 .4985
776 .4975
.66818 1.4966
860 .4957
902 .4947
944 .4938
.66986
.4928
.67028 1.4919
071 .4910
cos
I
cot
56°
tan
sin
;~~
.~..
.~~.
,:-,:':>
»:,...
,1.
.~.
.,i'.h
u,
.~
017
.83001
.82985
969
953
936
920
.82904
I
~.
~.1!-
20
19
18
17
16
15
14
13
12
11
10
034
.4900
I
21
'}~
6
5
4
3
2
I
0
~I
'.
,,~
,'iJ.(.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
cos
I
cot
.A
I
55°
35°
TABLE IV
cos
cot
1
1450 2350 3250
1460 2360 3260
34<>
tan
.55919 .67451 1.4826 .82904
887
943
493 .4816
871
968
536 .4807
855
.55992
578 .4798
839
.56016
620 .4788
.56040 .67663 1.4779 .82822
806
064
705 .4770
790
088
748 .4761
773
112
790 .4751
757
136
832 .4742
.56160 .67875 1.4733 .82741
724
184
917 .4724
208 .67960
708
.4715
692
232 .68002
.4705
256
675
045 .4696
.56280 .68088 1.4687 .82659
305
643
130 .4678
329
173 .4669
626
353
610
215 .4659
377
258 .4650
593
.56401 .68301 1.4641 . 82577
425
561
343 .4632
449
544
386 .4623
473
429 .4614
528
497
511
471 .4605
.56521 .68514 1.4596 .82495
478
545
557 .4586
569
600 .4577
462
593
446
642 .4568
617
685 .4559
429
.56641 .68728 1.4550 .82413
665
771 .4541
396
689
380
814 .4532
713,
363
857 .4523
736 I 900 .4514 I 347
.56760 1.68942 11.4505 ~82330
784 1.68985
314
.4496
808.69028
.4487
297
832
281
071 .4478
856
264
114 .4469
.56880 .69157 1.4460 .82248
904
200 .4451
231
928
214
243 .4442
952
286 .4433
198
.56976
329 .4424
181
.57000 .69372 1.4415 .82165
024
416 .4406
148
047
459 .4397
132
071
502 .4388
115
095
545 .4379
098
.57119 .69588 1.4370 .82082
143
631 .4361
065
167
675 .4352
048
191
718 .4344
032
215
761 .4335 .82015
.57238 .69804 1.4326 .81999
262
847 .4317
982
286
965
891 .4308
949
310
934 .4299
334 .69977
932
.4290
.57358 .7002111.42811.81915
I
,
3040
sin
'I
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22 I
676
660
645
.83629
613
597
581
565
.83549
533
517
501
485
.83469
453
437
421
405
.83389
373
356
340
324
.83308
292
276
260
244
.83228
212
195
179
163
.83147
131
115
098
082
.83066
050
I
113
I
cos
1.5399 .83867 60
851 59
.5389
835 58
.5379
819 57
.5369
.5359
804 56
1.5350 .83788 55
.5340
772 54
.5330
756 53
.5320
740 52
724 51
.5311
1.5301 .83708 50
692 49
.5291
53
750
155 .4891
54
775
197 .4882
55 .55799 .67239 1.4872
56
823
282 .4863
57
847
324 .4854
58
871
366 .4844
59
895
409 .4835
60 .55919 .67451 1.4826
I
sin
726
1240 2W
1230 21303030
I tan
tan
I
sin
I
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
I
sin
tan
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
.57358
381
405
429
453
.57477
501
524
548
572
.57596
619
643
667
691
.57715
738
762
786
810
.57833
857
881
904
928
.57952
976
.57999
.58023
.70021
064
107
151
194
.70238
281
325
368
412
.70455
499
542
586
629
.70673
717
760
804.
129
cos
I
cot
I
1. 4281
.4273
.4264
.4255
.4246
1.4237
.4229
.4220
.4211
.4202
1.4193
.4185
.4176
.4167
.4158
1. 4150
.4141
.4132
4124
'13976
.3968
.3959
.3951
.3942
1.3934
.3925
.3916
.3908
.3899
1.3891
.3882
.3874
.3865
.3857
1.3848
.3840
.3831
.3823
.3814
1.3806
.3798
.3789
.3781
.3772
1.3764
I
I
.4115
I
cos
.81915
899
882
865
848
.81832
815
798
782
765
.81748
731
714
698
681
.81664
647
631
614
597
.81580
563
546
530
513
.81496
479
462
445
428
.81412
395
378
361
1
344
181327
310
293
276
259
.81242
225
208
191
174
.81157
140
123
106
089
.81072
055
038
021
.81004
80987
970
953
936
919
.80902
848
.70891
935
.70979
.71023
066
.71110
154
198
242
I
047
285
.58070 .71329
094
373
417
118
461
1411
505
165
.58189 '.71549
212
593
637
236
260
681
725
283
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813
330
354
857
901
378
946
401
.58425 .71990
449 .72034
472
078
122
496
167
519
.58543 .72211
567
255
299
590
614
344
388
637
.58661 .72432
684
477
521
708
565
731
610
755
.58779 .72654
I
I'
I
1250 2150 3050
cot
1.4106
.4097
.4089
.4080
.4071
1.4063
.4054
.4045
.4037
.4028
1.4019
.4011
.4002
. 3994
1 3985
54°
tan
sin
I
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
I
'
1440 2340 3240
~
1260 2160 3060 360
sin
tan I cot
0 .58779 . 72654 1.3764
I
802
699 .3755
2
826
743 .3747
3
849
788 .3739
4
873
832 .3730
I
I
.80902
885
867
850
833
sin
60
59
58
57
56
6 .58896 .72877 1.3722 .80816 66
6
920
921 .3713
799 54
7
943 .72966 .3705
782 53
8
967 .73010 .3697
765 52
9 .58990
055 .3688
748 51
10 .59014 .73100 1.3680 .80730 50
11
037
144 .3672
713 49
12
13
14
061
084
108
189
234
278
.3663
.3655
.3647
696
679
662
48
47
46
15 .59131 .73323 1.3638 .80644 45
16
17
18
19
154
178
201
225
368
413
457
502
.3630
.3622
.3613
.3605
627
610
593
576
44
43
42
41
20 .59248 .73547 1.3597 .80558 40
21
272
592 .3588
541 39
22
295
637 .3580
524 38
23
318
681 .3572
507 37
24
342
726 .3564
489 36
25 .59365 .73771 1.3555 .80472 35
26
389
816 .3547
455 34
27
412
861 .3539
438 33
28
436
906 .3531
420
29
459
951 .3522
403
30 .59482 .73996 1.3514 .80386
31
506 74041 .3506
368
32
529
086 .3498
351
33
552
131 .3490
334
34
576
176 .3481
316
I
11
<4T~
Rn7QQ
35 59599 74711
I
~~36 622
37
646
38
669
39
693
40 .59716
41
739
42
763
43
786
44
809
45 .59832
856
46
47
879
48
902
49
926
267
312
357
402
.74447
492
538
583
628
.74674
719
764
810
32
31
30
29
28
27
26
?!;
I
855
.3465 I 282 24
.3457
264 23
.3449
247 22
.3440
230 21
1.3432 .80212 20
.3424
195 19
.3416
178 18
.3408
160 17
.3400
143 16
J. 3392 .80125 15
.3384
108 14
.3375
091 13
.3367
073 12
.3359
056
11
60 59949 .74900 1.3351 .80038 10
972
51
946 .3343
021 9
52 .59995 .74991 .3335' .80003 8
53 .60019 .75037
.3327
54
042
082 .3319
65 .60065 .75128 1.3311
56
089
173 .3303
57
112
219 .3295
.79986
968
.79951
934
916
135
264 .3287
899
581
59
158
310 .3278
881
60 .60182 .75355 1.3270 .79864
1
,-
C(\S
cot
~.~
I
tan
0
370
TABLE IV
cos
sin
7
6
5
4
3
2
I
0
'
tan
1270 2170 307°
cot
cos
0 .60182 .75355 1.3270
205
I
401 .3262
2
228
447 .3254
3
251
492 .3246
4
274
538 .3238
.
I
I.'
.79864
846
829
81 I
793
60
59
58
57
56
6 .60298 .75584 1.3230 .79776 66
6
321
629 .3222
758 54
7
344
675 .3214
741 53
8
367
721 .3206
723 52
9
390
767 .3198
706 51
10 .60414 .75812 1.3190 .79688 60
II
437
858 .3182
671 49
12
13
14
460
904
483
950
506 .75996
.3175
.3167
.3159
653
635
618
48
47
46
15 .60529 .76042 1.3151 .79600 45
16
17
18
19
553
576
599
622
088
134
180
226
20 .60645 .76272
21
668
318
22
691
364
23
714
410
24
738
4.56
25 .60761 .76502
26
784
548
27
807
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583
565
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530
36 :61 015 i:no 10 I :2985 I' 229
37
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057 .2977
21I
38
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103 .2970
193
39
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149 .2962
176
40 .61107 .J96
1.2954 .79158
41
130
242 .2946
140
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289 .2938
122
43
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335 .2931
105
44
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382 .2923
087
45 .61222 .77428 1.2915 .79069
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475 .2907
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47
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521 .2900
033
48
291
568 .2892 .79016
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314
615
44
43
42
41
1.3111 .79512 40
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1.3072 .79424 35
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28
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640 .3048
371
29
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353
30 .60876 .76733 1.3032 .79335
31
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318
32
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300
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t:r1aal .,U\~ I ,,,,,,
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. .~
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.78998
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.78891
873
855
58
520 .78035 .2815
837
59
543
082 .2807
819
60 .61566 .78129 1.2799 .78801
cos
cot
I
tan
I
sin
t.
24
23
22
21
20
19
18
17
16
16
14
13
12
a.
'.
.~
"
.~;::
12
13
.(.'
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14
if
'f.
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l.~'
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cot
cos
.78801
783
765
747
841
864
692
739
887
786
.2708
.2700
586
568
.~.
56
65
54
53
52
51
50
49
48
47
II
12
13
"r
I
'...,;'"
1...,'
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0
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.2247
14
16 .63271 .81703 1.2239
16
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752 .2232
17
316
800 .2225
18
338
849 .2218
48
49
660
402
683
.2437
450 .2430
52
751
594 .2408
53
774
642 .2401
54
796
690 .2393
65 .62819 .80738 1.2386
56
842
786 .2378
57
864
834 .2371
58
59
60
J!"L
A
.2268
.2261
.2254
46
41
40
39
38
37
36
35
34
934
916
861
843
824
.77806
788
769
887
882 .2364
751
909
930 .2356
733
62932 .80978 1.2349 .77715
cos I cot I tan
sin
.l:i I'
.I:.IJ.
iJ~
248
513
494
476
60
59
58
57
56
56
54
53
52
51
60
49
48
47
458 46
.77439
421
402
384
45
44
43
42
27
540
287 .2153
218
28
563
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199
29
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181
30 .63608 .82434 1.2131 .77162
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32
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125
33
675
580.
2109
107
34
698
629 .2102
088
35 .63720 j. 82678 11.2095 .77070
33
32
31
30
29
28
27
26
25
19
361
898 .2210
366 41
20 .63383 .81946 1.2203 .77347 40
21
406 .81995 .2196
329 39
22
428 .82044 .2189
310 38
23
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292 37
24
473
141 .2174
273 36
26 .63496 .82190 1.2167 .77255 35
26
518
238 .2160
236 34
33
32
31
30
29
28
27
26
25
I
36
742
727 .2088
37
765
776 .2081
38
787
825 .2074
39
810
874 .2066
40 .63832 .82923 1.20j9
854 .82972 .2052
41
42
877 .83022 .2045
43
899
071 .2038
44
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120 .2031
45 .63944 .83169 1.2024
46
966
218 .2017
47 .63989
268 .2009
12
48 .64011
051 24
033 23
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.76996 21
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959 19
940 18
921 17
903 16
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866 14
847 13
.2002
317
828
12
52
100
514 .1974
754
53
123
564 .1967
735
54
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613 .1960
717
56 .64167 .83662 1.1953 .76698
56
190
712 .1946
679
57
212
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661
8
7
6
6
4
3
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366 .1995
810 11
50 .64056 .83415 1.1988 .76791 10
078
51
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772 9
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60 .62706 .80498 1.2423 .77897 10
51
728
546 .2415
879 9
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r.r,
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510
558
606
45
44
43
42
388
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433
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39
456 .79972 .2504
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40 .62479 .80020 1.2497' .78079 20
41
502
067 .2489
061 19
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524
115 .2482
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43
547
163 .2475
025 17
44
570
211 .2467 .78007 16
45 .62592 .80258 1.2460 .77988 15
46
615
306 .2452
970 14
47
638
354 .2445
952 13
.,
180
203
225
550
11. 2534 1.78170
cos
.7771 5
696
678
660
4
022
171 .2320
641
5 .63045 .81220 1.2312 .77623
6
068
268 .2305
605
7
090
316 .2298
586
8
113
364 .2290
568
9
135
413 ,2283
550
10 ,63158 .81461 1.2276 .77531
.78532
514
496
478
~~.
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cot
0 .62932 .80978 1.2349
1
955 .81027
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2 .62977
075 .2334
3 . 63000
123 .2327
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27
401 .2594
315
28
206
449
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297
29
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279
30 .62251 .79544 1.2572 .78261
31
274
591 .2564
243
32
297
639.2557
225
I 206
33
320
686.2549
34
342
734 I .2542 I 188
J...
tan
sin
'
60
59
58
57
1290 2190 3090
15 .61909 .78834 1.2685
16
932
881 .2677
17
955
928 .2670
18 .61978 .78975
.2662
19 .62001 .79022 .2655
460
20 .62024 .79070 1.2647 .78442
]
21
046
17 .2640
424
22
069
164 .2632
405
23
092
212 .2624
387
24
115
259 .2617
369
25 .62138 .79306 1.2609 .78351
26
160
354 .2602
333
."
11
7
6
6
4
3
sin
4
658
316 .2769
729
6 .61681 .78363 1.2761 .78711
6
704
410 .2753
694
7
726
457 .2746
676
8
749
504 .2738
658
772
551 .2731
640
~9
10 .61795 .78598 1.2723 .78622
818 645 .2715 604
',t'
.
I
390
TABLE IV
0 .61566 .78129 1.2799
1
589
175 .2792
2
612
222 .2784
3
635
269 .2776
.~
32
31
30
29
28
27
26
..u
50 .61337 .77661 1.2876 .78980 10
51
360
708 .2869
962 9
52
383
754 .2861
944 8
53
406
801 .2853
54
429
848 .2846
55 .61451 .77895 1.2838
56
474
941 .2830
57
497 .77988
.2822
1280 2180 3080 380
8
7
6
5
4
3
2
1
0
58
234
811 .1932
642
59
256
860 .1925
623
60 .642791.83910 1.1918 .76604
I
,
1:-l1
cos
I
I
cot
n
tan
I
sin
2
1
0
'
.. ..11\.0 "\,,nn ftftftl'l
~
1300 2200 3100
I
I
0
1
2
3
4
5
6
7
8
9
10
II
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
5'1
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
I
1390
sin
400
tan
cos
.64279 .83910 1.1918 .76604
301.83960.
1910
586
323.84009.
1903
567
346
059. 1896
548
368
108 .1889
530
. 64390 .84158 1. 1882 . 76511
412
208 .1875
492
435
258.1868
473
457
307.1861
45,
479
357 .1854
436
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524
457 .1840
398
546
507. 1833
380
568
556 .1826
361
590
606 .1819
342
.646J2 .84656 1. 1812 .76323
63,
706 .1806
304
657
756.1799
286
679
806 .1792
267
70I
856. 1785
248
.64723 .84906 J. 1778 .76229
746 .84956
1771
210
768.85006.1764
192
790
057. 1757
173
812
107. 17,0
154
.64834 .85157 1.1743 .7613,
856
207. 1736
116
878
257.1729
097
901
308.1722
078
923
358. 1715
059
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967
458. 1702
022
.64989
509. 1695 .76003
.65011
559.1688.75984
Ujj
IbISI
965
26
.75946
927
908
889
870
.75851
832
813
794
775
.75756
738
719
700
680
.75661
642
623
604
585
.75566
547
528
509
490
.75471
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
I
0
'
I
cot
2290 3190
I
tan
49°
I
sin
I
'I
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
.65055!. 85660 1. 1674
077
710.1667
100
761 . 1660
122
811 .1653
144
862.1647
.65166 .85912 1.1640
188.85963.
1633
210 .86014 .1626
232
064. 1619
254
115 .1612
.65276 .86166 1.1606
298
216. 1599
320
267 .1592
342
318 .1585
364
368 .1578
.65386 .86419 1.1571
408
470 .156,
430
521 .1558
452
572.1551
474
623.1544
.65496 .86674 1.1538
518
725 .1531
540
776.1524
562
827. 1517
584
878. 1510
.65606 ,86929 1.1504
cos
bU~.
410
TABLE IV
cot
0
1
2
3
4
5
6
7
8
9
10
II
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35 .66371
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
.88732 11,1270
393
784. 1263
414
836 .1257
436
888 .1250
458
940.1243
.66480 .b8992 1.1237
501 .8904, .1230
523
097 .1224
54,
149 .1217
566
201 .1211
.66588 .89253 1.1204
610
306 .1197
632
358 .1191
653
410. 1184
675
463. 1178
.66697 .89515 1.1171
718
567 .1165
740
620.1158
762
672 .1152
783
725 .1145
.66805 .89777 1.1139
827
830. 1132
848
883.1126
870
935.1119
891.89988.1113
. 66913 .90040 1." 06
cos
132
1310221.3110
sin [tan
I cot I cas I
.65606 .86929 1. 1504 .75471 60
628.86980.
1497
452 59
6,0 .87031 .1490
433 58
672
082 .1483
414 57
694
133 .1477
39, 56
.65716 .87184 1.1470 .75375 55
738
236.1463
356 54
759
287.1456
337 53
781
338.14,0
318 52
803
389. 1443
299 51
.65825 .87441 1. 1436 .75280 50
847
492 .1430
261 49
869
543 .1423
241 48
891
59, .1416
222 47
913
646.14'10
203 46
.6593, .87698 1.1403 ,75184 45
956
749. 1396
16, 44
.65978
801 .1389
146 43
.66000
852 .1383
126 42
022
904 .1376
107 41
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066.88007.
1363
069 39
088
059. 1356
0,0 38
109
110. 1349
030 37
131
162. 1343 . 75011 36
.66153 .88214 1.1336 .74992 35
17,
265. 1329
973 34
197
317. 1323
953 33
218
369. 1316
934 32
240
421 . 1310
915 31
.66262 .88473 1.1303 .74896 30
284
524 .1296
876 29
857 28
306
576.1290
327
628. 1283 I 838 27
349
680.1276 I 81820
cot
1
48°
I
tan
I
-=
,1
'
1
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25
780
760
741
722
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683
664
644
625
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586
567
548
528
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489
470
451
431
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392
373
353
334
.74314
24
23~j
221;
21
20
19:J:!
18
17
16:Wt'
15
14
13
12
11
10
9
8
7
6
5
4
3
2
I
0
.~
..\>.
::Jf
\'
~~'
'
'
,
,
, " , ,.
iN
;,""
'.
:~,~
.f'",:':;
'
'
,
,
.S. -,.
~~.
~i-;t;i
~'
~
'~"
'
I'
1380 2280 318"
"
"I
,"
..
~
"
'I
.
',:...,
'
,.
.
,
sin
,
',
,
.";Jf:
.
"-,
_.
1320 2220 3120 420
sin
tan I cot
cos
0 . 66913 . 90040 1. 1106 .74314
295
1
935
093 .1100
276
2
956
146 .1093
3
978
199 .1087
256
4 . 66999
251 .1080
237
5 .6702J .90304 1. 1074 .74217
6
043
357 .1067
198
7
064
410 .1061
178
159
8
086
463 .1054
139
9
107
516 .1048
10 .67129 .90569 1. 1041 .74120
11
151
621 .1035
100
12
172
674 .1028
080
13
194
727 .1022
061
14
215
781 .1016
041
15 .67237 .90834 1. i009 .74022
16
258
887 .1003 .74002
17
280
940 .0996 .73983
963
18
301 .90993 .0990
944
19
323 .91046 .0983
20 .67344 .91099 1.0977 .73924
904
21
366
153 .0971
885
22
387
206 .0964
865
23
409
259 .0958
846
24
430
313 .0951
25.67452.91366
1.0945 .73826
806
26
473
419 .0939
787
27
495
473 .0932
767
28
516
526 .0926
747
29
538
580 .0919
30 .67559 .91633 1.0913 .73728
708
31
580
687 .0907
.0900
688
~~
~~~
~~O ~
34
64,.
847 .0888
649
35 .67666 1.9190 I 1.0881 .73629
610
36
688 .91955 .0875
590
37
709.92008
.0869
570
38
730
062 .0862
551
39
752
116 .0856
40 .67773 .92170 1.0850 .73531
41
795
224 .0843
511
491
42
816
277 .0837
472
43
837
331 .0831
44
859
385 .0824
452
45 .67880 .92439 1.0818 .73432
46
901
493 .0812
413
393
47
923
547 .0805
48
944
601 .0799
373
49
965
655 .0793
353
50 .67987 .92709 1.0786 .73333
51 .68008
763 .0780
314
52
029
817 .0774
294
53
051
872 .0768
274
54
072
926 .0761
254
55 .68093 .92980 1.0755 .73234
56
11'.93034
.0749
215
19,
57
136
088 .0742
175
58
157
143 .0736
155
59
179
197 .0730
60 .68200 .93252. 1.0724 , .73135
cas
I cot I tan I sin
1370 i27~ 3170
47°
TABLE IV
,
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
~~
34
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
,
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
430
sin
cot!
tan
cos
.68200 .93252' I .0724 .73135
221
306.0717
116
242
360.0711
096
264
415 .0705
076
285
469.0699
056
.68306 .93524 1.0692 .73036
327
578 .0686 .73016
349
633.0680.72996
370
688.0674
976
391
742.0668
957
.68412 .93797 1.0661 .72937
434
852.0655
917
455
906 .0649
897
476.93961
.0643
877
497.94016.0637
857
.68518 .94071 1.0630 .72837
539
125.0624
817
561
180. 0618
797
582
235 .0612
777
603
290 .0606
757
.68624 .94345 1.0599 .72737
645
400.0593
717
666
455.0587
697
688
510 .0581
677
709
565.0575
657
.68730 .94620 1.0569 .72637
751
676.0562
617
772
731 .0556
597
793
786.0550
577
814
841 .0544
557
.68835 .94896 1.0538 .72537
857.94952
.0532
517
~~~. 95~?~ . ~~~~
~~~
920:
118..0513
457
.689411.95173iI, 0507:.72437
962
229 .0501
417
.68983
284.0495
397
.69004
340.0489
377
025
395.0483
357
.69046 .95451 1.0477 .72337
067
506 .0470
317
088
562 .0464
297
109
618 .0458
277
130
673.0452
257
.69151 .95729 1. 0446 .72236
172
785 .0440
216
193
841 .0434
196
214
897.0428
176
235.95952.0422
156
.69256 .96008 1.0416 .72136
277
064.0410
116
298
120.0404
095
319
176 .0398
075
340
232.0392
055
.69361 .96288 1.0385 .72035
382
344.0379.72015
403
400.0373.71995
424
457 .0367
974
445
513 .0361
954
.69466 .96569 1. 0355 .71934
cos
133
1330 2230 3130
cot
tan
46°
I
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
~~
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
sin
1360 2260 3160
l
TABLE
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
n_~j433
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
1340 2240 314°
cot
tan
.69466
215
236
257
277
.70298
319
339
360
381
.70401
422
443
463
484
.70505
525
5'16
567
587
. 70608
628
649
670
690
.70711
I
TABLE V. RADIAN MEASURE,
COB
613
671
728
786
.98843
901
.98958
.99016
073
.99131
189
247
304
362
.99420
478
536
594
652
.99710
768
826
884
.99942
1.0000
cot
914
894
873
853
.71833
813
792
772
752
.71732
711
691
671
650
.71630
610
590
569
549
.71529
508
488
468
447
.71427
407
386
366
345
.71325
305
284
264
243 I
.0141
.0135
.0129
.0123
1.0117
.0111
.0105
.0099
.0094
1. 0088
.0082
.0076
.0070
. 0064
1.0058
.0052
.00'17
.0041
.0035
1. 0029
.0023
. 001 7
.0012
.0006
1.0000
.70998
978
957
937
.70916
896
875
855
834
.70813
793
772
752
731
.70711
tsn
si:1
I
450
203
182
162
141
.71121
100
080
059
039
.71019
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
"ft.
~{;l
f!;
-
4''''
...:;~.
~~t
".;r'!
"'.-,oj.
O. 00000 00 60° 1.0471976 1200
1 0.0174533
61 1 . 06465 08 121
2 0.03490 66 62 1 .08210 41 122
3 0.0523599
63 1.09955 74 123
4 0.06981 32 64 1. 11701 07 124
5 0.08726 65 65 1 . 1 3446 40 125
2.0943951
2. 11184 84
2.1293017
2.1467550
2. 16420 83
2.1816616
6 0.1047198 66 1.15191 73 126 2.1991149
7 0.1221730 67 1.1693706 127 2 . 21656 82
8 0.13962 63 68 1.1868239 128 2 . 23402 14
5
6
7
0.0014544
0.0017453
0.00203 62
5 0.0000242
6 0.0000291
7 0.00003 39
0 . 00008 24
O. 00008 73
1.48352 99
I .50098 32
1.5184364
I .53588 97
1.5533430
11.5707963
1.5882496
145
146
147
148
149
160
151
2.5307274
2.54818 07
2.56563 40
2.58308 73
2.6005406
2.6179939
2.6354472
0.00727 22
0.0075631
0.00785 40
0.0081449
0.0084358
0.00872 66
0.0090175
1 .30899
69
25
26
27
28
29
30
31
1154 '2.6878070
59
60
=
1330 2250 315°
..a.
L
1.0297443
1.0471976
118
119
120
25
26
27
28
29
30
31
134 '10.0098902134
0.6 1086 52
..'.
0.00523
0.00552
0.00581
0.00610
0.00639
O.00669
O.00698
8
9
10
11
12
13
14
15
16
51
17
60 18
69 19
78 20
87 21
95 22
04 23
13 24
O. 00006 30
0.0000679
0.00007 27
0.00007 76
0.0000921
O.00009 70
0.00010 18
0.0001067
0.00011 15
O. 000 I I 64
0.00012 12
0.0001261
0.0001309
0.0001357
0.0001406
0.0001454
0.0001503
29 152 2.6529005 32 0.0093084 32 0.0001551
~~t1.60570
+,6231562
34 10.59341 19 I 94,1.640609)
I~
0.0000048
0.0000097
0.0000145
0.0000194
O. 00494
85
86
87
88
89
90
91
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
I
1.32645 02
1.34390 35
1.36135 68
1.3788101
1.39626 34
1.4137167
1.4311700
1.44862 33
1.46607 66
0.43633 23
0.45378 56
0.47 I23 89
0.48869 22
0.5061455
0.5235988
0.5410521
38 0.66322 51
39 0.68067 84
0.6981317
0.7155850
42 0.73303 83
43 0.7504916
44 0.76794 49
15 0.78539 82
46 0.80285 15
47 0.82030 47
48 0.83775 80
49 0.85521 13
50 0.87266 46
51 0.8901179
52 0.9075712
53 0.92502 45
54 0.9424778
55 0.95993 11
56 0.9773844
57 0.99483 77
I 58 I . 01229 10
1
2
3
4
0.00003 88
0.00004 36
0.0000485
0.0000533
0.00005 82
25
26
27
28
29
30
31
96
97
0.0002909
f. 00058 18
0.00087 27
0.0011636
0.0023271
0.00261 80
0.00290 89
0.0031998
0.00349 07
0.0037815
0.00407 24
0.00436 33
0.0046542
1.20427 72
1.2217305
1.2391838
1.25663 71
1.27409 04
1.29154 36
0.62831 85
0.64577 18
0" 0.0000000
0'
1
2
3
4
8
9
10
11
2. 30383 46
12
2.3212879
13
2.3387412
14
15
2.3561945
2.37364 78 16
2.39110 11 17
2.40855 44 18
2.42600 77 19
2.4434610
20
2.46091 42 21
2.47836 75 22
2.49582 08 23
24
2.5132741
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
~I36
37
Seconds
O. 00000 00
2.2514747
2.26892 80
2.28638 13
0.1570796
0.1745329
0.1919862
0.2094395
0.22689 28
0.2443461
0.26179 94
0.27925 27
0.29670 60
0.3141593
0.33161 26
0.34906 59
0.36651 91
0.38397 24
0.4014257
0.4188790
~.~~~~n~
.
I
Minutes
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
32
26
24
23
22
21
20
19
18
17
16
15
14
13
12
1J
10
~
8
7
6
5
4
3
2
1
0
I
00
= 1.
0° TO 180°, RADIUS
Degrees
.96569 1.0355 .71934 60
487
625.0349
508
681 .0343
529
738.0337
549
794.0331
.69570 .96850 J.0325
591
907.0319
612 .96963 .0313
633.97020.0307
654
076.0301
.69675 .97133 1. 0295
696
189.0289
717
246.0283
737
302.0277
758
359.0271
.69779 .97416 1.0265
800
472 .0259
821
529 .0253
842
586.0247
862
643 .0241
.69883 .97700 1.0235
904
756.0230
925
813 .0224
946
870 .0218
966
927 .0212
.69987 .97984 1.0206
.70008 .98041 .0200
029
098 .0194
049
155.0188
070
213.0182
.70091 .98270 1.0176
112
327.0170
132
384.0164
153
441 .0158
-t741--4991
.0152
cos
134
440
IV
sin
1.67551 61 156
1 . 69296
94
1.7104227
1.72787 60
1.7453293
I .76278 25
1.78023 58
1.79768 91
I .81514 24
1.83259 57
1.85004 90
1.86750 23
1.88495 56
1.90240 89
1.9198622
1.93731 55
1.95476 88
1.9722221
1.98967 53
2.0071286
2.0245819
2.04203 52
157
158
159
60
161
162
163
164
165
166
167
168
169
170
171
172
173
174
75
176
177
2.05948 85 178
2.0769418 179
2.0943951 180
p""grppo::.
': 0 0001648
0.0001745
2.7227136
36
0.0104720
36
2.7401669
2.75762 02
2.77507 35
2.79252 68
2.8099801
2.8274334
2.84488 67
2.86234 00
2.8797933
2.89724 66
0.0107629
0.0110538
2.9321531
2. ~496064
2.96705 97
2.98451 30
3.0019663
3.01941 96
3.03687 29
3.0543262
3.0717795
3.08923 28
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
0.0116355
0.01192 64
0.01221 73
0.0125082
0.0127991
0.0130900
0.0133809
0.0136717
0.0139626
0.0142535
0.0145444
0.0148353
0.01512 62
0.01541 71
0.0157080
0.0159989
0.0162897
0.0165806
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
0.0001794
0.0001842
0.0001891
0.0001939
0.0001988
0.00020 36
0.00020 85
0.0002133
0.0002182
0.00022 30
0.00022 79
0.00023 27
0.0002376
0.00024 24
0.00024 73
0.0002521
0.00025 70
0.00026 18
0.00026 66
0.00027 15
0.00027 63
3.1066861
3.12413 94
3.14159 27
58
59
60
0.0168715
0.0171624
0.0174533
58
59
60
0.0002812
0.00028 60
0.01029 09
2 . 91469 99
I
O.01134 46
Minutes
I
Seconds
135
..
TABLE VI.
CONSTANTS
AND THEIR
LOGARITHMS.
Logarithm
Circumference
of a
Circumference
of a
Circumference
of a
Number of radians
Number of radians
Number of radians
N umber of degrees
Number of minutes
Number of seconds
11'= 3. 141 592653
circle in degrees. . . . . . . . . 360
=
circle in minutes
=21,600
circle in seconds. . . . . . . . . 1,296,000
=
in one degree
=0.017 4533
in one minute
=0.0002909
in one second
=0.0000048
in one radian. . . . . . . . . . . . 57.2957795
=
in one radian
=3437.7468
in one radian
=206,264.806
589793.
Also:
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
I
h = 6.2831853
h = 12.5663706
I
2'11'=
4
311'=
1
411'=
1
(;11'=
J
1r =
11'2
I V; J.7724539
=
1
=0.5641896
V;
M =0.4342945
Logarithm
0.798 1799
1.5707963
J .099 2099
0.1961199
4.1887902
0.622 0886
0.7853982
9.8950899-10
0.5235988
9.7189986-10
O. 318 3099
9.5028501-10
= 9.869 6044
0.994 2997
~
I e
1
I
e
v3
I
2.556 3025
4.334 4538
6. 112 6050
8.241 8774-10
6.4637261-10
4.6855749-10
1 .758
9.7514251-10
~.637 7843-10
0 . 362
=2.7182818
0.434 2945
=0.3678794
9.5657055-10
O. 150 5150
= 1.732 0508
0.238 5606
v'5 = 2.236 0680
0 . 349 4850
.-
1 mile per hour = 1.466 667 feet per second.
1 foot per second = 0.681 818 miles per hour.
I cu. ft. of water weighs 62.5 lb. = 1000oz. (approximat 'I.
1 gal. of water weighs 8t lb. (approximate).
1 gal. = 231 cu. in. (by law of Congress).
1 bu. = 2150.42 cu. in. (by law of Congress).
1 bu. = 1.2446 cu. ft. = £ cu. ft. (approximate).
1 cu. ft. = 7} gal. (approximate).
1 bbl. = 4.211 cu. ft. (approximate).
I meter = 39.37 inches (by law of Congress).
1 ft. = 30.4801 em.
1 kg. = 2.20462 lb.
1 gram = 15.432 grains.
1 lb. (av0irdupois)
1 lb.
1 liter
1 qt.
1 qt.
=
2157
2136
--
J lb. (avoirdupois)
1226
3.536 2739
5.3144251
0.497 1499
0.2485749
=2.3025851
I v2 = 1.414
453.592 4277 grams = 0.45359 kg.
I
~;
--+-~
-~
~
= 7000 grains (by law of Congress).
(apothecaries) = 5760 grains (by law of Congress).
= 1.05668 qt. (liquid) = 0.90808 qt. (dry).
(liquid) = 946.358 cc. = 0.946358 liters, or cu. dm.
(dry) = 1101.228 cc. = 1.101 228 liters, or cu. dm.
136
I
\
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